Steady-state input calculation for achieving a desired steady-state output of a linear systems

Steady-state input calculation for ac hieving a desired steady-state output of a linear systems Raffaele Romagnoli a Eman uele Garone a a Université Libr e de Bruxel les, B-1050 Bruxel les, Belgium (e-mail:rr omagno@ulb.ac.b e, e gar one@ulb.ac.b e). Abstract In this note we p ro vide an algorithm for the computation of the steady-state inpu t able t o achiev e the steady-state output trac k ing of any desired output signal representa b le as a rational transfer function. 1 In tro duction Output tracking problem is a well kno wn pro blem in the litera ture which has b een solved for the tra c king of steady-state output signals. One of the methods devel- op ed to so lve this problem is the internal mo del princi- ple [1]. The existence of the steady-state respo ns e needs the a symptotic stability of the considered system. Given a particular steady-state b ehavior of the output, and an asymptotically stable linear systems, it is possi- ble to find the co rresp onding steady-state input under the hypo thesis that the considered system is right in- vertible [2]. Then a feedforward controller that provides the ex a ct input sig nal to ac hieve the o utput tracking at steady-state can b e found. Despite the c omputation o f the steady state res ponse of a linear system is well kno wn in the litera tur e and can be found in every textb o ok (e.g. [3]), at the b est o f the author’s knowledge, the inv erse pr ocedure has no t been developed in the same ma nner , except for so me of the most common and simplest cases. The aim of this pape r is to provide an explicit in version algorithm, for comput- ing the steady-state input for the general case of r e fer- ence signa ls defined as fractional trasnfer functions. ⋆ This pap er w as not presented at an y IF AC meeting. Corresponding author Raffae le Romagnoli. E-mail: rro- magno@ulb.ac.be This w ork is p erformed in the framew ork of the BA TW AL pro ject financed by the W alloon region (Belgium). This research has been fund ed b y t he Mandats d’Impulsion Scienti fic "Optimization-free Control of Nonlinear Systems sub ject to Constrain ts" of the F onds de la Recherc h e Scien- tifique ( FN RS), Ref. F452617F . 2 Problem Stateme n t W e c onsider an Asymptocally Stable Linear Time In- v ariant SISO System in the form ˙ x ( t ) = Ax ( t ) + B u ( t ) y ( t ) = C x ( t ) + D u ( t ) (1) with y ( ⋅ ) ∈ I R q , u ( ⋅ ) ∈ I R p and x ( ⋅ ) ∈ I R n denoting the output, the input, and the sta te, resp ectiv ely . Its tra ns- fer function is denoted by W ( s ) = C ( sI − A ) − 1 B + D . Un- der the hypothesis of r igh t-inv ertibilit y [2] the assump- tion that the sys tem is SISO is without lo ss o f g ener- ality as the same results ar e directly applicable also to the MIMO case. F or this class of sy s tems we will show how to compute input signals ensuring the feedforward asymptotic tra c king of the following signals: (1) ˜ y d ( t ) = t k po lynomial signa l ; (2) ˜ y d ( t ) = Y d sin ( ω d t + ψ d ) sinusoidal sig nal ; (3) ˜ y d ( t ) = Y d e a d t with a d > 0 ex p onential signa l ; (4) ˜ y d ( t ) = e a d t Y d sin ( ω d t + ψ d ) pseudo-p erio dic sig na l; (5) ˜ y d ( t ) = t k e a d t sin ( ω d t + ψ d ) p olynomial pse udo - per iodic signal. 3 Input computatio n algorithms 3.1 Polynomial input s of or der k . T o find the the input u s ( t ) such tha t ˜ y d ( t ) = t k , we c o n- sider the asymptotic output resp onse to the p olynomial input t k that is ˜ y ( t ) = C 0 t k + C 1 t k − 1 + ⋯ + C k , (2) Preprin t submitted to Automatica Nov ember 9, 2018 where C i = 1 i !  d i W ( s ) ds i  s = 0 . (3) Giving as input the signa l u 0 ( t ) = 1 C 0 t k , (4) the r esulting asymptotic output is ˜ y ( t ) = t k + ˆ C 1 t k − 1 + ⋯ + ˆ C k (5) where ˆ C i = C i C 0 for i = 1 , ..., k . The idea is to find k inputs able to ca ncel the ˆ C i terms o f (5). This can b e done using the following a lgorithm ● Step 0 : u 0 ( t ) = t k C 0 , ˆ C i = C i C 0 for i = 1 , ..., k ; ● Step 1: for i = 1 ∶ k u i ( t ) = − ˆ C i C 0 t k − i ; for j = 1 ∶ k − i ˆ C i + 1 = ˆ C i + 1 − ˆ C i C 0 C j ; end end ● Step 2: u s ( t ) = ∑ k i = 0 u i ( t ) 3.2 Sinusoidal signal Considering the a symptotic output resp onse of a sinu- soidal input of the form U sin ( ω 0 t + ψ ) ˜ y ( t ) =  W ( j ω 0 ) U sin ( ω 0 t + ( ψ + arg ( G ( j ω 0 )))) (6) and defining as the des ired asymptotic resp onse ˜ y d ( t ) = Y d sin ( ω d t + ψ d ) (7) the input that gener ates (7) is u s ( t ) = U sin ( ω d t + ψ ) where U = Y d  W ( j ω d ) (8) ψ = ψ d − arg ( W ( j ω d )) (9) 3.3 Exp onential signal, with p ositive r e al ex p onent a > 0 The a s ymptotic resp onse of a n exp onential signal of the form u ( t ) = U e at is ˜ y ( t ) = W ( a ) U e at . (10) Hence, considering the desired output of the form ˜ y d ( t ) = Y d e a d t , it can b e o bta ined using u s ( t ) = U e a d t where U = Y d W ( a d ) (11) 3.4 Pseudo-p erio dic si gn als F ollowing the same pro cedure of the pr evious c a ses, we need to find the system resp onse of the following input u ( t ) = e at U sin ( ω t + ψ ) with a p ositive r eal n umber a > 0 . T o do so, we define the following tra nsfer function H ( s ) △ = L  w ( t ) e at  (12) where w ( t ) = L − 1 { W ( s )} , then the output resp onse is ˜ y ( t ) = e at  H ( j ω 0 ) U sin ( ω 0 t + ψ + arg ( H ( j ω 0 ))) . (13) Considering a s desire d function ˜ y d ( t ) = e a d t  H ( j ω d ) Y d sin ( ω d t + ψ d + arg ( H ( j ω d ))) , (14) the input able to trac k the desired o utput is u s ( t ) = e a d t U sin ( ω d t + ψ ) wher e U and ψ are computed using (8) a nd (9). 3.5 Polynomial pseudo-p erio dic signals In this last ca se all the previous signa ls are ta k en into account and the stea dy s ta te r e sponse of an input sig nal of the form u ( t ) = t k e at sin ( ω t + ψ ) with a pos itiv e real n um ber a is ˜ y ( t ) = e at        k  i = 0   k i   t k − i  H i ( j ω ) ⋅ sin ( ω t + ψ + arg ( H i ( j ω )))        , (15) where H i ( s ) △ = L  t i w ( t ) e at  , (16) and w ( t ) = L − 1 { W ( s )} . If the desired output ˜ y d ( t ) is expressed as (15), the propos ed a lgorithm fo r for p oly- nomial inputs (case 1) c a n b e rearra nged taking into ac- count the pseudo- perio dic sig nals to compute u s ( t ) . References [1] F rancis, Bruce A., and W. Murra y W onham. "The inter nal model principle of contro l theory ." Automatic a 12.5 (1976): 457-465. [2] Moylan, P . "Stable inv ersion of linear systems." IEEE T r ansactions on A utomatic Contr ol 22.1 (197 7): 74-78. [3] Go odwin, Graham C., Stefan F. Graebe, and Mario E. Salgado. "Cont rol system design." Upp er Sadd le River (2001): 13. 2