Robust feedback linearization

Transcription

1 Robust eedback linearization Hervé Guillard Henri Bourlès Laboratoire d Automatique des Arts et Métiers CNAM/ENSAM 21 rue Pinel Paris France {herveguillardhenribourles}@parisensamr Keywords: Nonlinear systems eedback linearization robust stability W-stability Abstract Classical eedback linearization has poor robustness properties and cannot be easily combined a H or H 2 type control law We propose here to transorm by eedback the original nonlinear system into its linear approximation around a given operating point and prove that this allows to preserve the good robustness properties obtained by a linear control law which it is associated This method constitutes a way o robustly controlling an uncertain nonlinear system around an operating point 1 Introduction In this paper we present a robust method or eedback linearization We consider a nonlinear system n states and m inputs described in a state-space orm by ẋ = (x) + m g i (x)u i = (x) + g(x)u (1) i=1 g(x) = [ g 1 (x) g m (x) ] u = [ u 1 u m ] T in which x R n denotes the state u R m is the control input and (x) g 1 (x) g m (x) are smooth vector ields deined on an open subset o R n We assume that the state is available or the control The main advantage o the classical eedback linearization method [1] comes rom the act that applying to system (1) an appropriate control law u c (x w) = α c (x) + β c (x)w the eedback system becomes linear rom new input w so it can be regulated using a classical linear eedback Still this method has two main drawbacks [2] Firstly since the system is clearly linearized by simpliying its nonlinearity this treatment can turn out not being robust i the nonlinearity is uncertain Moreover in many linear control laws (like or example H control or H 2 control) the required perormance is speciied by weighting (requentially or not) various variables As a consequence it is diicult to combine theses methods the classical linearization since the obtained system has no physical meaning: whatever the original nonlinear system is one usually comes down to the same Brunovsky orm (multiple chains o integrators) For example designing a weighting unction associated u in order to take into account some saturations will be extremely diicult because the design has to be done considering w which is not in reality applied to the system and the behaviour will be very dierent depending on whether one considers u or w so a robust design respect to w may be not robust at all respect to u In this paper we propose a linearization method which we believe can overcome these two diiculties For we use a undamental idea o automatic control: a eedback which modiies too much the natural behaviour o a system has little chance o being robust As a consequence eedback linearization must perorm on the original system the most little possible transormation Looked at rom that point o view a natural step is to use the nonlinear eedback which transorms the original system into its tangent linearized system around a given operating point Although this idea is natural its relevance has to be proved This is done in the sequel using an appropriate approach or stability and robustness namely W-stability [3] The paper will be organized as ollows In section 2 we recall some acts about W-stability In section 3 we propose a robust eedback linearization method whose properties are clariied in section 4 2 Review o W-stability In order to analyze the properties o the robust eedback linearizing method that will be proposed let us irstly recall some results rom [3] about W-stability This approach makes use o the Sobolev space W n o unctions h : R + R n such that h and its distributional derivative ḣ belong to

2 L n 2 The norm in W n is then deined as [ h W = h T (t)h(t)dt ḣ T (t)ḣ(t)dt ] 1 2 and the notion o local W-gain can be deined as usually [4] DEFINITION 1 : Let be G : W n W m a time-invariant nonlinear system and K = {k > 0 ɛ > 0 : Gu W k u W u W n such that u W < ɛ} I K is nonempty G is said to be locally-w-stable (l-w-s) and γ W l (G) = in(k) is called the local-w-gain o G Consider now that G : W n W m is a time-invariant nonlinear system described by state-space equations ẋ = F (x u) y = H(x u) equilibrium point (x u) = (0 0) Let G(s) = C(sI A) 1 B + D be the linear approximation o G around its equilibrium point ie A = F B = F 00 C = H D = H 00 u 00 u 00 The ollowing result illustrates an important property o Sobolev space W n PROPOSITION 1 : Assume that (A B) is stabilizable (C A) is detectable and G H Then G is l-w-s and γ W l (G) = G On this basis one can obtain a local version o the wellknown Small Gain Theorem u 1 + e 1 G 1 G 2 Figure 1 e THEOREM 1 : Consider the standard closed-loop system in Fig1 where G 1 : W n W m and G 2 : W m W n Assume that this closed-loop system is well-posed [5] ie there exist two operators H 1 and H 2 : W n+m W n+m such that e = H 1 u and y = H 2 u u2 Then i e = [ e1 e 2 ] [ y1 y = ] [ u1 and u = u 2 γ W l (G 1 )γ W l (G 2 ) < 1 ] the closed-loop system is l-w-s ie H 1 and H 2 are l-w-s This result provides a tool to analyze in an input-output ramework the robustness properties o a nonlinear closedloop system when small moves are considered Intuitively completeness o the small W -gain condition will guarantee stability respect to small signals Moreover although one loses the global aspect o the Small Gain Theorem this local version is less conservative since only local stability is concerned This will be used in section 4 to demonstrate the robustness properties o the eedback linearization method proposed in the next section 3 A robust method or eedback linearization Contrarily to the classical eedback linearization which transorms the original nonlinear system (1) into a Brunovsky orm the present method consists in transorming it into its tangent linearized system around an operating point here chosen as x = 0 that is ż = Az + Bv (2) A = x=0 and B = g(0) For suppose that distributions G 0 G 1 G n 1 deined as G 0 = span{g 1 g m } G 1 = span{g 1 g m ad g 1 ad g m } G i = span{ad k g j : 0 k i 1 j m} or i = 0 1 n 1 satisy the classical hypotheses (i) distribution G i has constant dimension near x = 0 or 0 i n 1 (ii) distribution G n 1 has dimension n (iii) distribution G i is involutive or 0 i n 2 Then as is well known [1] there exist real-valued unctions λ 1 (x) λ m (x) deined on a neighborhood U o x = 0 satisying or numbers r 1 r m such that r r m = n (j) or all i [1 m] all j [1 m] and all x U L gi λ j (x) = L gi L λ j (x) = = L gi L r j 2 λ j (x) = 0 (jj) the m m matrix L g1 L r 1 1 λ 1 (x) L gm L r1 1 λ 1 (x) M(x) = L g1 L r2 1 λ 2 (x) L gm L r 2 1 λ 2 (x) L g1 L rm 1 λ m (x) L gm L rm 1 λ m (x)

3 is nonsingular at x = 0 We will denote M = M(0) Consider on this basis the associated classical linearizing state eedback u c (x w) = α c (x) + β c (x)w (3) α c (x) = M 1 (x)n(x) β c (x) = M 1 (x) N(x) = [ L r 1 λ 1(x) and change o coordinates given by φ c (x) φ ci (x) λ 2(x) L rm λ m(x) ] T L r2 x c = φ c (x) (4) = [ φ c1 (x) φ cm (x) ] T [ = λ i (x) L λ i (x) L r i 1 λ i (x) Then one has the ollowing result ] T THEOREM 2 : Consider system (1) and suppose that (x) and g(x) satisy hypotheses (i) (ii) and (iii) Then under state eedback and change o coordinates deined by where α(x) β(x) φ(x) L = M α c u(x v) = α(x) + β(x)v z = φ(x) = α c (x) + β c (x)lt 1 φ c (x) = β c (x)r 1 (5) = T 1 φ c (x) T = φ c x=0 and R = M 1 x=0 system (1) is transormed into system (2) Proo: Applying classical linearizing eedback (3) together change o coordinates (4) it is known rom [1] that original nonlinear system (1) is transormed into where A c = x c = A c x c + B c w (6) A c1 0 r1 r 2 0 r1 r 3 0 r1 r m 0 r2 r 1 A c2 0 r2 r 3 0 r2 r m 0 rm r 1 0 rm r 2 0 rm r 3 A cm each r i r i matrix A ci being equal to A ci = and B c1 0 r1 1 0 r1 1 0 r1 1 O r2 1 B c2 0 r2 1 0 r2 1 B c = 0 rm 1 0 rm 1 0 rm 1 B cm each r i 1 matrix B ci being equal to B ci = [ ] T On the other hand (A B) being a controllable pair it is well known rom classical linear control theory (see [6] or example) that there exist matrices T (n n) L (m n) and R (m m) such that T (A BRL) T 1 = A c and T BR = B c T and R being non singular On this basis it is immediate to see that system (6) is transormed into (2) using eedback and change o coordinates w = LT 1 x c + R 1 v (7) z = T 1 x c (8) Combining (3)(4) (7)(8) it is clear that eedback u = α c (x) + β c (x)w = α c (x) + β c (x)lt 1 φ c (x) + β c (x)r 1 v = α(x) + β(x)v (9) together change o coordinates z = φ(x) = T 1 φ c (x) (10) transorms original nonlinear system (1) into its tangent linearized version (2) around x = 0 Moreover linear parts o (9) and (10) obviously satisy α = 0 x=0 φ x=0 = I n n and β(0) = I m m since these eedback and change o coordinates have no inluence on the linear part o original nonlinear system (1) As a consequence one has T = φ c L = M α c and R = M 1 x=0 x=0 which concludes the proo

4 4 Robustness analysis o the method P Figure 2 As was said above we claim that this method has certain robustness properties which can be analyzed using the concept o W-stability Consider in Fig2 as in [7] the standard diagram or robustness analysis o an uncertain system where R p 2 represents the measured signals in our context supposed to be state x R n o nonlinear time-invariant system P (ie p 2 = n) R m 2 is the control signal and R p 1 R m1 are introduced to depict the uncertainty operating on the original system is nonlinear time-invariant and such that γ Wl ( ) δ A = C = h x=0 x=0 B 1 = g 1 (0) B 2 = g 2 (0) D 1 = k 1 (0) and D 2 = k 2 (0) Suppose that one has ound a state eedback linear controller K (no matter how it has been constructed) which stabilizes P (s) and such that the transer unction rom to ie F l (P K) satisies F l (P (s) K) < 1 δ I this is the case the closed-loop system in Fig4 is stable or all uncertainty l satisying (s) δ (s) P (s) This system can be represented in a state-space orm by ẋ = (x) + g 1 (x) + g 2 (x) = h(x) + k 1 (x) + k 2 (x) (11) = x K Suppose that we are interested in robustly controlling this uncertain system around a given operating point x = x chosen here or convenience as x = 0 A possible way to deal this problem makes use o the linearizing eedback law derived in section 2 Figure 4 Returning now to the uncertain nonlinear system o Fig2 let us associate the linearizing eedback o Theorem 2 (s) P (s) Figure 3 For let us consider in Fig3 the corresponding linear problem where P (s) is the transer unction o the linear approximation o P around x = 0 ie P (s) : ẋ = Ax + B 1 + B 2 = Cx + D 1 + D 2 = x (12) LF : = α(x) + β(x)v 2 z = φ(x) where α(x) β(x) and φ(x) are constructed according to Theorem 2 on the basis o (x) and g 2 (x) to linear controller K that is v 2 = Kz The system relying to = z is now linear However considering the general closed-loop system in Fig5 Λ : ( ) ( ) is still nonlinear and given by ż = [ ] φ Az + B 2 + g 1(x) x=φ 1 (z) = [h(x) + k 2 (x)α(x)] x=φ 1 (z) + [k 1(x)] x=φ 1 (z) y 2 = z + [k 2 (x)β(x)] x=φ 1 (z) v 2 (13)

5 by (5) I linear controller K stabilizes P (s) deined in (12) and satisies F l (P (s) K) < 1 δ (15) then T(Λ K) is locally W -stable and v 2 LF P K Figure 5 = x φ(x) Λ = z A crucial issue concerns the point at which one evaluates robustness (see [8]) Indeed one is interested in obtaining robustness or system P : ( ) ( ) controlled by K and LF (Fig 61) but linear controller K which has been designed to this aim (at least around x = 0 since it has been computed on the basis o P (s)) is now acting on Λ (Fig62) and this situation can deteriorate robustness properties that have been obtained However the key point is that Λ and P have in act the same linear approximation around the operating point x = 0 thanks to the particular linearizing eedback used here γ W l (T(Λ K)) < 1 δ As a consequence the present eedback ensures local- W-stability or every nonlinear uncertainties such that γ W l ( ) δ Proo: Firstly Let T (s) be the linear approximation o T(Λ K) around z = 0 Considering state equations (14) A + B 2 K stable applying Proposition 1 leads to Then since clearly γ Wl (T(Λ K)) = T (s) T (s) = F l (Λ(s) K) where Λ(s) is the linear approximation o Λ around x = 0 one has γ Wl (T(Λ K)) = F l (Λ(s) K) (16) On the other hand Λ(s) is deined as Λ(s) : ż = Āx + B 1 + B 2 = Cx + D 1 + D 2 = z P LF+K v 2 Λ K y 2 the dierent matrices being computed on the basis o (13) Because o α φ x=0 = 0 x=0 = φ 1 = I n n and β(0) = I m m z z=0 one has Ā = A B1 = [ ] φ g 1(x) = B 1 B2 = B 2 x=φ 1 (0) Figure 61 Figure 62 Let us deine as T(Λ K) the nonlinear operator relying to in Fig62 According to state-space equations (13) o Λ T(Λ K) is given by [ ] φ ż = (A + B 2 K)z + g 1(x) x=φ 1 (z) = [h(x) + k 2 (x)α(x)] x=φ 1 (z) + [k 1(x)] x=φ 1 (z) + [k 2 (x)β(x)] x=φ 1 (z) K z (14) Then one has the ollowing result THEOREM 3: Consider the closed-loop system in Fig5 where P is given by equations (11) Suppose that (x) and g 2 (x) satisy the hypotheses o Theorem 2 and that linearizing eedback LF and change o coordinates φ(x) are given [ h C = + k 2 α(x) + k 2(x) α ] φ 1 z z=0 = C x=φ 1 (0) D 1 = D 1 and D 2 = [k 2 (x)β(x)] x=φ 1 (0) = D 2 thus proving that Λ(s) = P (s) Then or any linear controller K achieving (15) one has remembering (16) γ Wl (T(Λ K)) = F l (P (s) K) < 1 δ As a consequence according to Theorem 1 the closedloop interconnection is l-w-stable or every nonlinear uncertainties such that γ W l ( ) δ

6 Remark 2: This method which coners to the eedback linearized system the behaviour o the original nonlinear system or small moves around x = 0 (here supposed to be the design point) will lose its robustness properties or systems having very strong nonlinearities since the linear behaviour imposed by the linearized eedback law will be too constraining Still as ar as nonlinear robust control around an equilibrium point is concerned consider that the linear controller K achieves the optimal value o attenuation (which is not oten desirable but is here supposed or clarity o discussion) ie F l (P (s) K) < 1 δ max then the linearizing eedback proposed here is the best one that can be associated this linear controller since it does not deteriorate the closed-loop system robustness properties (and clearly no linearizing eedback can improve them) Moreover it can ensure in any case stability or an exact model all over the operating domain and as a consequence is clearly better than the linear controller alone (this is shown in [10] in the particular case o a steam turbine) Remark 3: There exist an ininite number o pairs (linear controller-linearizing eedback) achieving the same result In particular there exists a linear controller K c to be associated the classical linearizing eedback ie K c = (R 1 K L)T 1 but practically this is not a convenient way or designing controllers since a physical analysis is required in H 2 or H methods Remark 4: The method described here or robustly controlling uncertain systems must be compared nonlinear H control ([9][7]) A theoretical comparison is not easy Still a possible dierence could concern the size o the signals considered since W-stability (see Deinition 1) allows to analyse only small moves o a nonlinear system However the proposed method does not involve the numerical problems o nonlinear H control associated Hamilton- Jacobi equations or NLMI s Remark 5: We have given in this paper a general theoretical ramework to ideas that had been applied success on the particular example o a steam turbine ([2][10]) Remark 6: The method developed here or state eedback linearizable systems can be applied in an input-output linearization context to asymptotically minimum phase nonlinear systems but the details need to be worked out this design point Using then W-stability it is proved that this is the only eedback that does not deteriorate the robustness properties obtained by the linear control law Reerences [1] A Isidori Nonlinear control systems Springer-Verlag Heidelberg 1989 [2] H Bourlès F Colledani and MP Houry Robust continuous speed governor control or small-signal and transient stability IEEE Trans Power Syst vol PS-12 pp [3] H Bourlès and F Colledani W-stability and local input-output stability results IEEE Trans Automat Contr vol AC-40 pp [4] C A Desoer and M Vidyasagar Feedback Systems: Input-Output Properties New York: Academic 1975 [5] J C Willems The Analysis o Feedback Systems Cambridge MA: MIT Press 1971 [6] T Kailath Linear systems Prentice Hall Englewood Clis NJ 1980 [7] W-M Lu and JC Doyle Robustness analysis and synthesis or nonlinear uncertain systems IEEE Trans Automat Contr 42 pp [8] J C Doyle and G Stein Robustness observers IEEE Trans Automat Contr 24 pp [9] A Isidori and A Astoli Disturbance attenuation and H control via measurement eedback in nonlinear systems IEEE Trans Automat Contr 37 pp [10] F Colledani Régulation coordonnée tension/vitesse des groupes turbo-alternateurs PHd thesis Paris-Sud Orsay Conclusion In this paper we have proposed a method or robustly controlling around a design point a nonlinear eedback linearizable system P This method consists in associating a robust linear control law to a particular linearizing eedback This eedback instead o resulting in the classical Brunovsky orm is computed in order that the eedback linearized system coincides P (s) the tangent linearized system o P around