Model-free control of non-minimum phase systems and switched systems

Mo del-free con trol of non-minim um phase systems and switc hed systems Lo ¨ ıc MICHEL GR ´ EI - D ´ ep artement de g ´ enie ´ ele ctrique et g ´ enie informatique Universit ´ e du Qu ´ eb e c ` a T r ois-R ivi` er es C.P. 500, T r ois-R ivi` er es, G9A 5H7, Canada, QC Abstract This brief 1 presen ts a simple deriv ation of the standard mo del-free control for the non-minim um phase systems. The robustness of the prop osed metho d is studied in sim ulation considering the case of switc hed systems. 1 This w ork is distributed under CC license http://creativecommons.org/licenses/ by- nc- sa/3.0/ 1 In tro duction The model-free con trol metho dology , originally proposed b y [1], has b een widely suc- cessfully applied to many mec hanical and electrical pro cesses. The model-free control pro vides go o d p erformances in disturbances rejection and an efficient robustness to the pro cess internal changes. The con trol of non-minim um phase systems has b een deeply studied and successful metho ds ha v e b een prop osed (e.g. [2] [3] [4] [5] [6] [7] [8] [9]). Since the mo del-free con trol can not a priori stabilize a non-minim um phase system [1], we prop ose a p ossible deriv ation of the original mo del-free con trol law, dedicated to the control of non-minimum phase systems. The dynamic performances are esp ecially tested in the case of switched systems. The pap er is structured as follo ws. Section I I presents an ov erview of the mo del- free con trol metho dology including its adv an tages in comparison with classical metho d- ologies. Section II I discusses the application of the modified mo del-free con trol, called NM-mo del-free con trol, for non-minim um phase systems. Some concluding remarks ma y be found in Section IV. 2 Mo del-free con trol: a brief o v erview 2.1 General principles 2.1.1 The ultra-lo cal mo del W e only assume that the plant b eha vior is well appro ximated in its op erational range b y a system of ordinary differen tial equations, which migh t b e highly nonlinear and time-v arying. ∗ The system, whic h is SISO, ma y b e therefore described b y the input- output equation E ( t, y , ˙ y , . . . , y ( ι ) , u, ˙ u, . . . , u ( κ ) ) = 0 (1) where • u and y are the input and output v ariables, • E , which might b e unkno wn, is assumed to be a sufficien tly smooth function of its arguments. Assume that for some in teger n , 0 < n ≤ ι , ∂ E ∂ y ( n ) 6≡ 0. F rom the implicit function theorem we ma y write lo cally y ( n ) = E ( t, y , ˙ y , . . . , y ( n − 1) , y ( n +1) , . . . , y ( ι ) , u, ˙ u, . . . , u ( κ ) ) By setting E = F + α u w e obtain the ultr a-lo c al mo del. ∗ See [1, 10] for further details. 1 Definition 2.1 [1] If u and y ar e r esp e ctively the variables of input and output of a system to b e c ontr ol le d, then this system admits the ultr a-lo c al mo del define d by: y ( n ) = F + α u (2) wher e • α ∈ R is a non-physical c onstant p ar ameter, such that F and αu ar e of the same magnitude; • the numeric al value of F , which c ontains the whole “structur al information ”, is determine d thanks to the know le dge of u , α , and of the estimate of the derivative y ( n ) . In all the n umerous kno wn examples it w as possible to set n = 1 or 2. 2.1.2 Numerical v alue of α Let us emphasize that one only needs to give an appro ximate numerical v alue to α . It would b e meaningless to refer to a precise v alue of this parameter. 2.2 In telligen t PI con trollers 2.2.1 Generalities Definition 2.2 [1] we close the lo op via the in telligen t PI controller , or i-PI c on- tr ol ler, u = − F α + ˙ y ( n ) ∗ α + C ( ε ) (3) wher e • y ∗ is the output r efer enc e tr aje ctory, which is determine d via the rules of flatness- b ase d c ontr ol ([11, 12]); • e = y ∗ − y is the tr acking err or; • C ( ε ) is of the form K P ε + K I R ε . K P , K I ar e the usual tuning gains. Equation (3) is c al le d mo del-fr e e c ontr ol law or mo del-fr e e law. The i-PI controller 3 is compensating the p oorly kno wn term F . Con trolling the system therefore b oils down to the control of a precise and elementary pure in tegrator. The tuning of the gains K P and K I b ecomes therefore quite straigh tforw ard. 2 2.2.2 Classic con trollers See [13] for a comparison with classic PI controllers. 2.3 A first academic example: a stable monov ariable linear system In tro duce as in [1, 10] the stable transfer function ( s + 2) 2 ( s + 1) 3 (4) 2.3.1 A classic PID controller W e apply the well kno wn metho d due to Bro ¨ ıda (see, e.g. , [14]) by appro ximating System 4 via the following dela y system K e − τ s ( T s + 1) K = 4, T = 2 . 018, τ = 0 . 2424 are obtained thanks to graphical tec hniques. The gain of the PID con troller are then deduced [14]: K P = 100(0 . 4 τ + T ) 120 K τ = 1 . 8181, K I = 1 1 . 33 K τ = 0 . 7754, K D = 0 . 35 T K = 0 . 1766. 2.3.2 i-PI. W e are employing ˙ y = F + u and the i-PI controller u = − [ F ] e + ˙ y ? + C ( ε ) where • [ F ] e = [ ˙ y ] e − u , • y ? is a reference tra jectory , • ε = y ? − y , • C ( ε ) is an usual PI controller. 2.3.3 Numerical sim ulations Figure 1(a) sho ws that the i-PI con troller b eha ves only sligh tly better than the classic PID con troller (Fig. 1(b)). When taking into account on the other hand the age- ing pro cess and some fault accommo dation there is a dramatic change of situation: Figure 1(c) indicates a clear cut sup eriorit y of our i-PI controller if the ageing pro- cess corresp onds to a shift of the p ole from 1 to 1 . 5, and if the previous graphical iden tification is not rep eated (Fig. 1(d)). 3 2.3.4 Some consequences • It migh t b e useless to introduce dela y systems of the type T ( s ) e − Ls , T ∈ R ( s ) , L ≥ 0 (5) for tuning classic PID con trollers, as often done today in spite of the quite in volv ed iden tification pro cedure. • This example demonstrates also that the usual mathematical criteria for robust con trol become to a large irrelev an t. • As also sho wn by this example some fault accommodation may also be achiev ed without having recourse to a general theory of diagnosis. (a) i-PI control (b) PID control (c) i-PI control (d) PID control Figure 1: Stable linear monov ariable system (Output (–); reference (- -); denoised output (. .)). 4 3 Con trol of non-minim um phase systems W e explain in this section, ho w to deriv e the mo del-free con trol la w (3) in order to stabilize and guarantee certain p erformances for non-minimum phase systems. W e will sho w that the prop osed con trol law is also robust to disturbances and switc hed mo dels. 3.1 Discrete mo del-free con trol la w for non-minim um phase systems Firstly , consider the discretized mo del-free con trol law, whic h is typically used for a digital implementation. Definition 3.1 [15] F or any discr ete moment t k , k ∈ N , one defines the discr ete c ontr ol ler i-PI. u k = u k − 1 − 1 α  y ( n )   k − 1 − y ( n ) ∗   k  + C ( ε ) | k (6) wher e • y ∗ is the output r efer enc e tr aje ctory; • ε = y ∗ − y is the tr acking err or; • C is a usual c orr e ctor PI wher e K P , K I ar e the usual tuning gains. The discr ete intel ligent c ontr ol ler is also c al le d discr ete mo del-fr e e c ontr ol law or discr ete mo del-fr e e law. Non-minim um phase systems are characterized b y negativ e zero(s). Suc h zero can b e appro ximated by a delay since e − T s ≈ 1 − T s using a T a ylor expansion. T o comp ensate the effect of the dela y , that ma y destabilize the control, w e take the deriv ative of the output y instead of the output y to create the measuremen t feedbac k. This w ay allows to an ticipate the v ariations of y and finally cancel the disturbances asso ciated to the presence of the delay . W e define consequently the i*-PI con troller for non-minimum phase systems. Definition 3.2 F or any discr ete moment t k , k ∈ N , one defines the discr ete c on- tr ol ler i*-PI for non-minimum phase systems. λ and δ j ar e r e al c o efficients. u k = G ( ε ) ( u k − 1 − n X j =1 δ j  λ y ( j )   k − 1 − y ( j ) ∗   k  ) (7) wher e 5 • y ∗ is the output r efer enc e tr aje ctory; • ε = y ∗ − y is the tr acking err or; • G ( ε ) is c al le d a gain function and is either a pur e gain or an inte gr ator † . The discr ete intel ligent c ontr ol ler is also c al le d discr ete NM-mo del-fr e e c ontr ol law or discr ete NM-mo del-fr e e law. Practically , sim ulations sho w that n = 2 is sufficien t ‡ to ensure at least the stabilit y of the model-free con trol closed-lo op. Therefore, (7) is written : u k = G ( ε )  u k − 1 − δ 2  λ d 2 y d t 2     k − 1 − d 2 y ∗ d t 2     k  − δ 1  λ d y d t     k − 1 − d y ∗ d t     k  (8) F or the following applications, w e c ho ose the gain function as an in tegrator, with a K i constan t, suc h that : G ( ε ) = K i Z t 0 ε d t (9) 3.2 Applications Consider the systems Σ 1 , Σ 2 , Σ 3 and Σ 4 , whic h are minim um and non-minim um phase systems, and whic h are described resp ectiv ely b y the state-space representations : Σ 1 :=    ˙ x =  0 − 1000 100000 − 5000  x +  2 . 10 4 0  u y =  − 10 1  x (10) Σ 2 :=    ˙ x =  0 − 900 80000 − 3500  x +  2 . 10 4 0  u y =  − 13 1  x (11) Σ 3 :=    ˙ x =  0 − 900 80000 − 3500  x +  2 . 10 4 0  u y =  +13 1  x (12) † Dep ending on the application, a pure gain can b e enough to ensure go o d tracking p erformances. ‡ The p ossibilit y of reducing n will b e studied in a future w ork. 6 Σ 4 :=    ˙ x =  0 − 400 70000 − 1500  x +  2 . 10 4 0  u y =  +5 1  x (13) The unitary step resp onse of these systems is presented Fig. 2. Figure 2: Step responses of the systems Σ 1 , Σ 2 and Σ 4 . The follo wing figures present some preliminary examples of the application of the i*-PI control. Figure 3 presen ts the trac king of an exp onen tial reference for the system Σ 1 (with a fo cus on the b eginning of the transien t). Figures 4 and 5 sho w the resp onse y of the controlled system when resp ectiv ely Σ 1 switc hes to Σ 2 and with the addition of a sin usoidal disturbance on the v ariable u . Figures 6 and 7 presen t the control of switc hed systems; in particular the commutation from a non-minim um phase system to a minimum phase system. Figures 8 and 9 present the trac king of a sin usoidal reference when systems switch. The case where Σ 1 switc hes to Σ 3 has b een already studied in [16]. W e inv estigated the application of the mo del-free con trol in a microgrid environmen t under load / transfer function c hanges. These c hanges imply substan tial modifications of the controlled mo dels. 7 (a) T ransient with stabilization. (b) F o cus on the b eginning of the transient. Figure 3: T rac king of an exp onen tial reference. 8 Figure 4: T rac king of an exp onen tial reference; Σ 1 switc hes to Σ 2 at t = 5 ms. Figure 5: T rac king of an exp onen tial reference with a sin usoidal disturbance added on u suc h that ˜ u = 5 cos( 2 π 5 . 10 − 5 t ). 9 Figure 6: T rac king of an exp onen tial reference; Σ 1 switc hes to Σ 3 at t = 5 ms. Figure 7: T rac king of an exp onen tial reference; Σ 1 switc hes to Σ 4 at t = 1 . 5 ms. 10 Figure 8: T rac king of a sinusoidal reference; Σ 1 switc hes to Σ 2 at t = 3 ms. Figure 9: T rac king of a sinusoidal reference; Σ 1 switc hes to Σ 3 at t = 3 ms. 4 Concluding remarks W e presented some simulation results that confirm the fact that the NM-mo del-free con trol or i*-PI controller, designed for the con trol of non-minim um phase systems, 11 ensure go od tracking p erformances. W e ev aluated the p erformances in presence of disturbances and in the case of switched systems. In particular, the NM-mo del-free con trol is able a priori to control b oth minimum and non-minimum-phase systems. The prop osed con trol law seems to hav e the same prop erties than the original mo del- free con trol [1] for whic h its p erformances hav e b een successfully pro ved in sim ulation when controlling switched systems (e.g. [16]). F urther w ork will concern the study of the stability of the NM-mo del-free con trol method and its applications to netw ork ed systems. References [1] M. Fliess and C. 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