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ROBUST FEEDBACK LINEARIZATION WITHOUT FULL STATE INFORMATION
ROBUST FEEDBACK LINEARIZATION WITHOUT FULL STATE INFORMATION Ana L´ ucia Driemeyer Franco ∗,1 Henri Bourl` es ∗ Edson Roberto De Pieri ∗∗
∗ Laboratoire SATIE / ENS Cachan 61 Av. du Pr´esident Wilson, 94230, Cachan, France ∗∗ DAS / CTC / Universidade Federal de Santa Catarina Campus Trindade, 88040-900, Florian´ opolis, Brazil
Abstract: Robust feedback linearization of nonlinear systems without full state information is studied in this paper. The state is reconstructed from the output and the input by a linear observer combined with a diffeomorphism. It is shown that a separation principle holds, i.e., the estimated state can be used for performing robust feedback linearization, which exactly transforms the nonlinear system into a linear system equal to its linear approximation around a nominal operating point. Local stability of the resulting closed loop is proved by theoretical arguments and c IFAC 2007 its robustness is illustrated through an example. Copyright Keywords: Nonlinear systems, Feedback linearization, Robust control, Observers, Magnetic Bearings
1. INTRODUCTION In control theory, linearization of nonlinear systems is largely used due to the fact that for linear systems the choice of control techniques is wider and the design is more systematic. In the simplest case, a linear controller is calculated for the linear approximation of a nonlinear system around an operating point and directly applied to the plant. This method generally works only in a small neighborhood of such a point, i.e. in the region where the linear approximation is valid. When the system state is far from this point, the linear controller does not behave as desired. To avoid this problem, feedback linearization is used. With this method, the nonlinear system (in the nominal case) is exactly transformed into a linear system by feedback and diffeomorphism, after what a linear controller is applied. Hence, such a linearization is valid in the whole region where the 1
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diffeomorphism is defined, which yields a better behavior of the controller even far from the operating point. With classical feedback linearization (Isidori, 1989), the resulting linearized system is in the Brunovsky canonical form, a non-robust form extremely sensitive to system parameter variations. With the so-called robust feedback linearization, Guillard and Bourl`es (2000) have eliminated this drawback, by obtaining a linearized system that corresponds to the linear approximation of the nonlinear system around a nominal operating point. In this case, only a little transformation is performed on the original system, thus the properties of the linear design, including robustness, still hold locally for the nonlinear closed loop. The application of the three aforementioned methods was fully explored in Franco et al. (2006) and the clear advantage of robust feedback linearization was pointed out. In the same paper, robustness (regarding model uncertainties) of the performance obtained with robust feedback linea-
rization associated with a McFarlane-Glover H∞ controller (McFarlane and Glover, 1992) was demonstrated. However, feedback linearization presents one difficulty: full state information is needed in order to linearize the system. The objective of this paper is to study the case where only the system output is known. This can be dealt with by cancelling the nonlinear dynamics, as proposed by Isidori and Ruberti (1984). Nevertheless, this is not a suitable solution in many situations, because the application of this approach is restricted to systems with stable zero dynamics and also because the cancellation induces a lack of robustness. A more appropriate solution, investigated in this paper, is the use of an observer to reconstruct the state information. In the literature, many nonlinear observers are proposed, specially high-gain observers (Esfandiari and Khalil, 1992; Atassi and Khalil, 1999). In those papers, the control law (calculated independently from the observer) is supposed to be bounded. This hypothesis can not be verified when using any feedback linearization associated with a linear (hence unbounded) control input. In this context, since the estimation is performed in a linearization loop, a natural choice appears to be having a linear observer, which estimates the state of the linearized system. However, this is not straightforward. The linearized system can only be obtained by feedback linearization, requiring a previous knowledge of the full state. Here, it is demonstrated that this approach can be implemented, despite the possible vicious circle described above, provided that robust feedback linearization is used. This is also illustrated through an application example. The paper is organized as follows. In Section 2, the concept of feedback linearizable system and the robust feedback linearization method are briefly reviewed. In Section 3, the relation between feedback linearizable systems and observable ones is established. A method to perform robust feedback linearization without full state information by using a linear observer is proposed in Section 4. The application of this method to a magnetic bearing system, presented in Section 5, illustrates the theory. This paper concerns nonlinear MIMO systems, affine in the input.
2. ROBUST FEEDBACK LINEARIZATION Consider the nonlinear system with n states and m inputs described by the state-space equation x˙ = f (x) + g(x)u = f (x) +
m X
gi (x)ui
(1)
i=1
where x ∈ Rn denotes the state, u ∈ Rm is the control input and f (x), g1 (x), · · · , gm (x) are smooth vector fields defined on an open subset of
Rn . The equilibrium (x0 , u0 ) is chosen, without loss of generality, as (x0 , u0 ) = (0, 0). The following result is classical (see, e.g., Isidori, 1989). Lemma 1. If the nonlinear system (1) satisfies the well-known conditions for feedback linearization, then there exist m real-valued scalar functions λi (x) defined on a P neighborhood U of x0 , with m relative degrees ri , i=1 ri = n, such that: (a) for all i ∈ [1, m], all j ∈ [1, m] and all x ∈ U r −2
Lgi L0f λj (x) = · · · = Lgi Lfj
λj (x) = 0
(2)
(b) the decoupling matrix M (x) given by Lg1 Lrf1 −1λ1(x) · · · Lgm Lrf1 −1λ1(x) .. .. .. M(x)= (3) . . . Lg1Lrfm −1λm(x) · · · LgmLrfm −1λm(x)
is invertible for all x ∈ U, and (c) the vector φ(x) defined as
λi (x) φ1 (x) Lf λi (x) φ(x) = ... , φi (x) = .. . φm (x) Lrfi −1 λi (x)
(4)
is a local diffeomorphism in U.
Accordingly to Lemma 1, the nonlinear system x˙ = f (x) + g(x)u T y = λ(x) = λ1 (x) · · · λm (x)
(5)
is said to be a feedback linearizable system. The following result is proved by Guillard and Bourl`es (2000). Lemma 2. If the state x is known, the robust feedback linearization of the nonlinear system (5) is accomplished by using the diffeomorphism xr = φr (x)
(6)
and the feedback linearizing control law ur (x, v) = αr (x) + βr (x)vr
(7)
where vr is the new input, φr (x) = T −1 φ(x), αr (x) = α(x) + β(x)LT −1 φ(x), βr (x) = β(x)R−1 , L = −M (0)∂x α(0), T = ∂x φ(0), R = M −1 (0), T α(x) = −M −1 (x) Lrf1 λ1 (x) · · · Lrfm λm (x) and β(x) = M −1 (x). The values of L, T and R are chosen such that ∂x αr (0) = 0, βr (0) = Im and ∂x φr (0) = In . The linearized system is obtained in the form x˙ r = Ar xr + Br vr (8) y = Cr xr with Ar = ∂x f (0), Br = g(0) and Cr = ∂x λ(0), which corresponds to the linear approximation of the nonlinear system (5) around (x0 , u0 ) = (0, 0).
In the next sections, the case when the state x of the system (1) is not known is studied. 3. OBSERVABILITY The concepts of feedback linearizable system and observable system are connected, if the measured output is appropriate. This is shown in the following theorem. Theorem 1. Consider system (1). If the conditions for feedback linearization are satisfied and if it is possible to obtain measurable functions ¯ 1 (x), · · · , λ ¯ m (x) for which the Lemma 1 holds λ true, then (i) the nonlinear system x˙ = f (x) + g(x)u (9) ¯ 1 (x) · · · λ ¯m (x) T ¯ y = λ(x) = λ ¯ where y = λ(x) is the measured output, is both feedback linearizable and observable, and (ii) the robust feedback linearized system x˙ r = Ar xr + Br vr y = C¯r xr
(10)
¯ with C¯r = ∂x λ(0), is observable, that is, the pair (C¯r , Ar ) is observable. Proof: (i) System (9) is feedback linearizable since it is assumed that (1) satisfies the conditions for feedback linearization. To analyze the observability of system (9), the observability map, proposed by Zeitz (1984) (see also, e.g., Maggiore and Passino, 2003), is used. Roughly speaking, the observability map is constructed by differentiating each function of the output y until the order equal to its relative degree minus 1. This map is invertible if, and only if, the state x may be written uniquely as a function of y, u and their respective derivatives, which means that the system is observable. ¯ i (x) satisfy For system (9), since the functions λ Lemma 1(a), the observability map does not depend on the input u nor its derivatives and is given by ¯ λi (x) ω1 (x) ¯i (x) Lf λ ω(x) = ... , ωi (x) = (11) .. . ωm (x) ri −1 ¯ L λi (x) f
From Lemma 1(c), ω(x) corresponds to the local diffeomorphism φ(x). Consequently, the observability map is invertible in U (since a diffeomorphism is always invertible). Therefore, system (9) is observable.
(ii) Calculating ∂x ω(0) yields ¯ 1 (0) ∂x λ c¯r1 ¯ 1 (0)∂x f (0) c¯r1 Ar ∂x λ .. .. . . ¯1 (0) (∂x f (0))r1 −1 c¯r Ar1 −1 ∂x λ 1 r .. .. ∂x ω(0) = = . . ¯ ∂x λm (0) c¯rm ¯ ∂x λm (0)∂x f (0) c¯rm Ar .. .. . . rm −1 rm −1 ¯ c¯rm Ar ∂x λm (0) (∂x f (0)) (12) where c¯r1 , . . . , c¯rm are the rows of matrix C¯r . Since ω(x) corresponds to the local diffeomorphism φ(x), it is known that ω −1 (x) exists and both ω(x) and ω −1 (x) are differentiable. By the inverse function theorem, ∂x ω(x) is nonsingular and, in particular, that ∂x ω(0) is nonsingular. Hence, ∂x ω(0) corresponds to the n linearly independent rows of the observability matrix of the linearized system (10), which proves that the observability matrix has rank n. Therefore, the pair (C¯r , Ar ) is observable.
4. ROBUST FEEDBACK LINEARIZATION WITHOUT FULL STATE INFORMATION When all state variables of a nonlinear system are known, it is possible to use the robust feedback linearization given in Lemma 2 to linearize that system. When the state is not known, it is clearly impossible to do it. But, if the nonlinear system satisfies the conditions in Theorem 1, then one can use an observer to obtain an estimation of the state and replace this estimation in the control feedback law (7). If a linear observer is used, then an estimation of the state x ˆr of the linearized system is obtained. The estimation of the state x ˆ of the original nonlinear system is calculated using the “inverse” diffeomorphism xˆ = φ−1 xr ). r (ˆ This modified feedback linearizing loop is shown in Figure 1. vr-
ur-
αr (ˆ x) + βr (ˆ x)vr
Nonlinear System
y-
6 x ˆ
Linear Observer φ−1 xr ) r (ˆ
x ˆr
Fig. 1. Feedback linearizing loop with observer. As said in the introduction, it is not clear whether the linear observer works or not, due to the possible vicious circle mentioned there. The next theorem proves that the robust feedback linearization
fˆ(xr , vr , ǫr ) = θr ◦ σr f ◦ σr + θr ◦ σr g ◦ σr αr ◦ σ ˆr ◦ η + θr ◦ σr g ◦ σr βr ◦ σ ˆr ◦ η vr fˆ(xr , vr , ǫr ) = fˆ(0, 0, 0) + ∂xr fˆ(0, 0, 0)xr + ∂vr fˆ(0, 0, 0)vr + ∂ǫr fˆ(0, 0, 0)ǫr + O(x2r , vr2 , ǫ2r )
makes it possible to use such observer, since the estimation error converges to zero. In other words, it is proved that a separation principle holds. Theorem 2. Consider the nonlinear system x˙ = f (x) + g(x)u ¯ y = λ(x)
(15)
¯ 0 ) = 0. If the nonlinear system (15) with y0 = λ(x satisfies the conditions in Theorem 1 and vr is a stabilizing control for the linearized system (10), then x ˆ˙ r = Ar x ˆr + Br vr + Lobs (y − C¯r xˆr )
(16)
combined with the diffeomorphism xˆ = φ−1 xr ) , σ ˆr (ˆ xr ) r (ˆ
(17)
is a locally exponentially stable observer for system (15) controlled by the robust linearizing feedback ur (ˆ x, vr ) = αr (ˆ x) + βr (ˆ x)vr
(18)
where αr (ˆ x), βr (ˆ x) and φr (ˆ x) are defined according to Lemma 2. Furthermore, the estimation error defined as ǫr = xr − x ˆr
(19)
converges locally exponentially to zero. Proof: Applying the control law u = ur (ˆ x, vr ) to system (15) yields x˙ = f (x) + g(x) αr (ˆ x) + βr (ˆ x)vr (20) Since xr = φr (x), it follows that
x˙ r = ∂x φr (x)x˙ , θ(x)x˙
(21)
Replacing x˙ from (20) in this expression yields x˙ r = θ(x) f (x) + g(x) αr (ˆ x) + βr (ˆ x)vr (22) The state x may be written as x=
φ−1 r (xr )
, σr (xr )
(23)
while the estimated state x ˆ is given by (17). From (19), x ˆr = xr − ǫr , η(xr , ǫr )
(24)
Replacing (17), (23) and (24) in (22), it is possible to rewrite that equation in terms of xr , vr and ǫr , x˙ r = fˆ(xr , vr , ǫr )
(25)
with fˆ(xr , vr , ǫr ) given in (13) and fˆ(0, 0, 0) = 0.
(13) (14)
The function in (13) may be also written as in (14), where O(x2r , vr2 , ǫ2r ) represents the high-order terms of the Taylor expansion about (0, 0, 0). In order to calculate the partial derivatives of fˆ(xr , vr , ǫr ) with respect to xr , vr and ǫr about the fixed point (0,0,0), the following special properties of the functions η, σr , σ ˆr , θr , f , g, αr and βr are used: f (0) = 0 g(0) = Br αr (0) = 0 βr (0) = Im σr (0) = 0 σ ˆr (0) = 0 θr (0) = In
(26) (27) (28) (29) (30) (31) (32)
η(0, 0) = 0 ∂xr σr (0) = In ∂xˆr σ ˆr (0) = In ∂x f (0) = Ar ∂x αr (0) = 0 ∂xr η(0, 0) = In ∂ǫr η(0, 0) = −In
(33) (34) (35) (36) (37) (38) (39)
Using the chain rule to calculate the derivatives of composed functions and applying properties (26)(39), one obtains after some algebra ∂ fˆ ∂f ∂σr (0, 0, 0) = (0) (0) ∂xr ∂x ∂xr (40) ∂αr ∂σ ˆr ∂η + Br (0) (0) (0, 0) = Ar ∂x ∂x ˆr ∂xr ∂ fˆ (0, 0, 0) = Br (41) ∂vr ∂ fˆ ∂αr ∂σ ˆr ∂η (0, 0, 0) = Br (0) (0) (0, 0) = 0 (42) ∂ǫr ∂x ∂x ˆr ∂ǫr Replacing (40), (41) and (42) in (14) and neglecting the high-order terms O(x2r , vr2 , ǫ2r ) yields fˆ(xr , vr , ǫr ) = Ar xr + Br vr
(43)
and it follows from (25) that x˙ r = Ar xr + Br vr The estimation error dynamics is given by ǫ˙r = x˙ r − x ˆ˙ r
(44)
(45)
Using (16) and (44), the estimation error dynamics (45) may be rewritten as ǫ˙r = (Ar − Lobs C¯r )ǫr (46) ¯ From Theorem 1(ii), the pair (Cr , Ar ) is observable. Therefore, it is possible to choose a gain Lobs such that all eigenvalues of (Ar − Lobs C¯r ) lie in the left-half of the complex plane. Hence, the estimation error (19) converges locally exponentially to zero and the linear observer (16) combined to the diffeomorphism (17) is a locally exponentially stable observer for the nonlinear system (15) controlled by the robust linearizing feedback (18).
Remark 1. The statement of the above theorem is not valid for the classical feedback linearization. If an analogous procedure is made with the Brunovsky canonical form, it is not possible to ensure that the estimation error converges to zero. Properties (26)-(39) play a crucial role for this convergence, especially properties (28), (29), (32) and (37), which characterize the robust feedback linearization as seen in Lemma 2.
which yields a good performance/robustness tradeoff, as seen in the simulations. The estimation of the state x ˆ is obtained from the diffeomorphism x ˆr1 x ˆr2 x ˆr3 x ˆ = r 8I 2 x ˆ 2I (ˆ x −ˆ x ) (ˆ x +I )2 0
(k+2ˆ xr1 )
r4
k2
r3
−
0
k3
r1
+
r3
0
(k−2ˆ xr1 )2
−I0
(50)
5. APPLICATION EXAMPLE Consider the magnetic bearing system described by Queiroz and Dawson (1996) (see also Franco et al., 2006). Defining x1 as the rotor position, x2 as the rotor velocity, x3 and x4 as the currents, T u = u1 u2 as the input voltages, this system may be put in the state-space form x2 0 0 L0 (x3 +I0 )2 − (x4 +I0 )2 m (k−2x1 )2 (k+2x1 )2 0 0 u x˙ = R1 (k−2x1 )x3 2x2 (x3 +I0 ) + k−2x1 − 0 − L L k−2x 0 0 1 k+2x R2 (k+2x1 )x4 2x2 (x4 +I0 ) 1 0 − + k+2x1 L0 L0 T y = x1 x3 (47) where m is the rotor mass, I0 is the premagnetization current, R1 and R2 are the resistances of the circuits and L0 and k are positive constants depending on the system construction. Their nominal values are m = 2 kg, k = 2.0125 × 10−3 m, L0 = 3 × 10−4 Hm, R1 = R2 = 1 Ω and I0 = 2 × 10−2 A. The equilibrium point is x0 = 0. System (47) satisfies the conditions of Theorem 1, thus it is feedback linearizable and observable. The robust (47) is 0 8L0 I02 mk3 x˙ r = 0 0 10 0 y= 00 1
feedback linearized form of system 1 0 −2I0 k 2I0 k
0 x 0 r
0
0
2L0 I0 −2L0 I0 mk2 mk2 xr −kR1 0 L0 −kR2 0 L0
0 0 0 0 + Lk 0 vr 0 0 Lk0
(48)
By Theorem 1, system (48) is observable. Observer: A linear observer in the form (16) is designed for system (48). Since the zeros of this system are at infinity, rapid eigenvalues can be chosen for the matrix (Ar − Lobs C¯r ). For the set of eigenvalues {−36, −40, −44, −32}, the observer gain matrix obtained with the robust pole assignment algorithm from Matlab place (Kautsky et al., 1985) is T 44.0 370.6 −91.1 −2736.6 Lobs = (49) −3.1 −156.2 94.5 719.6
Robust Feedback Linearization: According to Lemma 2, αr (ˆ x) and βr (ˆ x) are calculated using α(ˆ x), β(ˆ x), φ(ˆ x), L, T and R given below. ˆ2 (ˆ x3 +I0 ) 0x R1 xˆ3 + 2L(k−2ˆ 2 x1 ) α(ˆ x) = (51) 2 2ˆ x2 L0 (k+2ˆ x1 )(ˆ 3 +I0 ) R2 x ˆ4 + (k−2ˆx1 )3 (ˆx4x+I ) 0 β(ˆ x) =
"
L0 k−2ˆ x1 −m(k+2ˆ x1 ) L0 (k+2ˆ x1 )(ˆ x3 +I0 ) 2(ˆ x4 +I0 ) (k−2ˆ x1 )2 (ˆ x4 +I0 )
0
#
x ˆ1 x ˆ2
φ(ˆ x) = L0 (ˆx3 +I0 )2 (ˆ x4 +I0 )2 m (k−2ˆx1 )2 − (k+2ˆx1 )2 x ˆ3 " # 0 R1 2I0 R2 0 0 −2I mk mk L= −kR1 0 0 −2I 0 k L0 1 0 0 0 0 1 0 0 T = 8L0 I02 2L0 I0 −2L0 I0 mk3 0 mk2 2 mk 0 0 1 0 " # L0 0 R = −mk Lk0 2I0
(52)
(53)
(54)
(55)
(56)
k
H∞ Controller: Since the state of system (48) is estimated, a controller vr = Kr x ˆr designed for the system with transfer matrix Gr (s) = (sI − Ar )−1 Br may be used. The H∞ method with loopshaping proposed by McFarlane and Glover (1992) allows to calculate the controller Kr analytically. For the loop-shaping, the weighting matrices W1 and W2 are chosen such that the frequency response of W2 Gr W1 (and consequently, the frequency response of Kr Gr ) is appropriate. Here are chosen W1 = I and 500(s + 2) W2 = diag , 95, 35, 15 (57) s The weighting matrix W2 adds an integrator to the first row of W2 Gr W1 , which is related to the rotor position x1 , to avoid steady-state errors, and a zero to better shape the position response. To the other lines of W2 Gr W1 , related to the velocity x2 and the currents x3 and x4 , only gains are added. A value γr ≈ 4 is used to calculate the controller. This yields a robustness index of 25%.
Simulations: A comparison with the linear controller Kr (combined with the linear observer) directly applied to the nonlinear system is presented below, to illustrate the performance of the closed loop with robust feedback linearization without full state information. Parameter variations are included, to evaluate the robustness of the method as well. The simulations are carried out with Simulink/Matlab, using the Dormand-Prince algorithm with a maximal step size of 0.1 ms. For these simulations, it is supposed that the parameter variations may be ±10% for m, L0 , R1 and R2 and ±5% for k, yielding 32 different combinations of the extreme values, which are all tested. The initial condition for the position x1 is 0.55mm, which allows the evaluation of the behavior of the controllers far from the equilibrium point. The results for (i) the closed loop with the linear control directly applied to the nonlinear system and (ii) the closed loop with the robust feedback linearization without full state information are given in Fig. 2 and 3 respectively. Linear Control Nominal Parameter Variation
Position x1 (mm)
0.8 0.6 0.4 0.2 0 −0.2 −0.4
0
0.5
1
1.5
2 Time (s)
2.5
3
3.5
4
Fig. 2. Rotor position x1 for (i). Robust Feedback Linearization without Full State Information Nominal Parameter Variation
Position x1 (mm)
0.8 0.6 0.4 0.2 0 −0.2 −0.4
0
0.5
1
1.5
2 Time (s)
2.5
3
3.5
4
Fig. 3. Rotor position x1 for (ii). The closed loop (ii) behaves as desired, having a good nominal performance, and for all the considered parameter variations the response remains close to the nominal one. The closed loop (i) presents a bad nominal performance and is unstable for 5 combinations of the parameter variations.
6. CONCLUDING REMARKS As shown by the theory in Section 4 and illustrated by the simulations in Section 5, it is possible to
perform the robust feedback linearization without full system information of a nonlinear system whose state is unknown. In particular, Theorem 2 proves that the state of the nonlinear system can be reconstructed with a linear observer and a diffeomorphism (provided that robust feedback linearization is used), which is a kind of separation principle. The same does not hold true when classical feedback linearization is used, as explained in Remark 1. Results connecting the concepts of feedback linearizable system and observable system are given in Section 3. In future works, the robustness of the approach (indicated by the simulations results) and the optimal choice of the observer gain matrix will be further studied.
REFERENCES Atassi, Ahmad N. and Hassan K. Khalil (1999). A separation principle for the stabilization of a class of nonlinear systems. IEEE Trans. Autom. Control 44(9), 1672–1687. Esfandiari, F. and H. K. Khalil (1992). Output feedback stabilization of fully linearizable systems. Int. J. Control 56, 1007–1037. Franco, A. L. D., H. Bourl`es, E. R. De Pieri and H. Guillard (2006). Robust nonlinear control associating robust feedback linearization and H∞ control. IEEE Trans. Autom. Control 51(7), 1200–1207. Guillard, H. and H. Bourl`es (2000). Robust feedback linearization. In: Proc. 14th Int. Symp. Mathematical Theory of Networks and Systems (MTNS’2000). Perpignan, France. Isidori, A. (1989). Nonlinear Control Systems - An Introduction. Springer-Verlag. Isidori, A. and A. Ruberti (1984). On the synthesis of linear input-output responses for nonlinear systems. Syst. Control Lett. 4, 17–22. Kautsky, J., N. K. Nichols and P. Van Dooren (1985). Robust pole assignment in linear state feedback. Int. J. Control 41, 1129–1155. Maggiore, M. and K. M. Passino (2003). A separation principle for a class of non-UCO systems. IEEE Trans. Autom. Control 48(7), 1122– 1133. McFarlane, D. and K. Glover (1992). A loop shaping design procedure using H-infinity synthesis. IEEE Trans. Autom. Control 37(6), 759–769. Queiroz, M. S. and D. M. Dawson (1996). Nonlinear control of active magnetic bearings: a backstepping approach. IEEE Trans. Control Syst. Technol. 4(5), 545–552. Zeitz, M. (1984). Observability canonical (phasevariable) form for nonlinear time-variable systems. Int. J. Syst. Science 15(9), 949–958.
∗ Laboratoire SATIE / ENS Cachan 61 Av. du Pr´esident Wilson, 94230, Cachan, France ∗∗ DAS / CTC / Universidade Federal de Santa Catarina Campus Trindade, 88040-900, Florian´ opolis, Brazil
Abstract: Robust feedback linearization of nonlinear systems without full state information is studied in this paper. The state is reconstructed from the output and the input by a linear observer combined with a diffeomorphism. It is shown that a separation principle holds, i.e., the estimated state can be used for performing robust feedback linearization, which exactly transforms the nonlinear system into a linear system equal to its linear approximation around a nominal operating point. Local stability of the resulting closed loop is proved by theoretical arguments and c IFAC 2007 its robustness is illustrated through an example. Copyright Keywords: Nonlinear systems, Feedback linearization, Robust control, Observers, Magnetic Bearings
1. INTRODUCTION In control theory, linearization of nonlinear systems is largely used due to the fact that for linear systems the choice of control techniques is wider and the design is more systematic. In the simplest case, a linear controller is calculated for the linear approximation of a nonlinear system around an operating point and directly applied to the plant. This method generally works only in a small neighborhood of such a point, i.e. in the region where the linear approximation is valid. When the system state is far from this point, the linear controller does not behave as desired. To avoid this problem, feedback linearization is used. With this method, the nonlinear system (in the nominal case) is exactly transformed into a linear system by feedback and diffeomorphism, after what a linear controller is applied. Hence, such a linearization is valid in the whole region where the 1
Supported by CAPES/COFECUB, contract 489/05.
diffeomorphism is defined, which yields a better behavior of the controller even far from the operating point. With classical feedback linearization (Isidori, 1989), the resulting linearized system is in the Brunovsky canonical form, a non-robust form extremely sensitive to system parameter variations. With the so-called robust feedback linearization, Guillard and Bourl`es (2000) have eliminated this drawback, by obtaining a linearized system that corresponds to the linear approximation of the nonlinear system around a nominal operating point. In this case, only a little transformation is performed on the original system, thus the properties of the linear design, including robustness, still hold locally for the nonlinear closed loop. The application of the three aforementioned methods was fully explored in Franco et al. (2006) and the clear advantage of robust feedback linearization was pointed out. In the same paper, robustness (regarding model uncertainties) of the performance obtained with robust feedback linea-
rization associated with a McFarlane-Glover H∞ controller (McFarlane and Glover, 1992) was demonstrated. However, feedback linearization presents one difficulty: full state information is needed in order to linearize the system. The objective of this paper is to study the case where only the system output is known. This can be dealt with by cancelling the nonlinear dynamics, as proposed by Isidori and Ruberti (1984). Nevertheless, this is not a suitable solution in many situations, because the application of this approach is restricted to systems with stable zero dynamics and also because the cancellation induces a lack of robustness. A more appropriate solution, investigated in this paper, is the use of an observer to reconstruct the state information. In the literature, many nonlinear observers are proposed, specially high-gain observers (Esfandiari and Khalil, 1992; Atassi and Khalil, 1999). In those papers, the control law (calculated independently from the observer) is supposed to be bounded. This hypothesis can not be verified when using any feedback linearization associated with a linear (hence unbounded) control input. In this context, since the estimation is performed in a linearization loop, a natural choice appears to be having a linear observer, which estimates the state of the linearized system. However, this is not straightforward. The linearized system can only be obtained by feedback linearization, requiring a previous knowledge of the full state. Here, it is demonstrated that this approach can be implemented, despite the possible vicious circle described above, provided that robust feedback linearization is used. This is also illustrated through an application example. The paper is organized as follows. In Section 2, the concept of feedback linearizable system and the robust feedback linearization method are briefly reviewed. In Section 3, the relation between feedback linearizable systems and observable ones is established. A method to perform robust feedback linearization without full state information by using a linear observer is proposed in Section 4. The application of this method to a magnetic bearing system, presented in Section 5, illustrates the theory. This paper concerns nonlinear MIMO systems, affine in the input.
2. ROBUST FEEDBACK LINEARIZATION Consider the nonlinear system with n states and m inputs described by the state-space equation x˙ = f (x) + g(x)u = f (x) +
m X
gi (x)ui
(1)
i=1
where x ∈ Rn denotes the state, u ∈ Rm is the control input and f (x), g1 (x), · · · , gm (x) are smooth vector fields defined on an open subset of
Rn . The equilibrium (x0 , u0 ) is chosen, without loss of generality, as (x0 , u0 ) = (0, 0). The following result is classical (see, e.g., Isidori, 1989). Lemma 1. If the nonlinear system (1) satisfies the well-known conditions for feedback linearization, then there exist m real-valued scalar functions λi (x) defined on a P neighborhood U of x0 , with m relative degrees ri , i=1 ri = n, such that: (a) for all i ∈ [1, m], all j ∈ [1, m] and all x ∈ U r −2
Lgi L0f λj (x) = · · · = Lgi Lfj
λj (x) = 0
(2)
(b) the decoupling matrix M (x) given by Lg1 Lrf1 −1λ1(x) · · · Lgm Lrf1 −1λ1(x) .. .. .. M(x)= (3) . . . Lg1Lrfm −1λm(x) · · · LgmLrfm −1λm(x)
is invertible for all x ∈ U, and (c) the vector φ(x) defined as
λi (x) φ1 (x) Lf λi (x) φ(x) = ... , φi (x) = .. . φm (x) Lrfi −1 λi (x)
(4)
is a local diffeomorphism in U.
Accordingly to Lemma 1, the nonlinear system x˙ = f (x) + g(x)u T y = λ(x) = λ1 (x) · · · λm (x)
(5)
is said to be a feedback linearizable system. The following result is proved by Guillard and Bourl`es (2000). Lemma 2. If the state x is known, the robust feedback linearization of the nonlinear system (5) is accomplished by using the diffeomorphism xr = φr (x)
(6)
and the feedback linearizing control law ur (x, v) = αr (x) + βr (x)vr
(7)
where vr is the new input, φr (x) = T −1 φ(x), αr (x) = α(x) + β(x)LT −1 φ(x), βr (x) = β(x)R−1 , L = −M (0)∂x α(0), T = ∂x φ(0), R = M −1 (0), T α(x) = −M −1 (x) Lrf1 λ1 (x) · · · Lrfm λm (x) and β(x) = M −1 (x). The values of L, T and R are chosen such that ∂x αr (0) = 0, βr (0) = Im and ∂x φr (0) = In . The linearized system is obtained in the form x˙ r = Ar xr + Br vr (8) y = Cr xr with Ar = ∂x f (0), Br = g(0) and Cr = ∂x λ(0), which corresponds to the linear approximation of the nonlinear system (5) around (x0 , u0 ) = (0, 0).
In the next sections, the case when the state x of the system (1) is not known is studied. 3. OBSERVABILITY The concepts of feedback linearizable system and observable system are connected, if the measured output is appropriate. This is shown in the following theorem. Theorem 1. Consider system (1). If the conditions for feedback linearization are satisfied and if it is possible to obtain measurable functions ¯ 1 (x), · · · , λ ¯ m (x) for which the Lemma 1 holds λ true, then (i) the nonlinear system x˙ = f (x) + g(x)u (9) ¯ 1 (x) · · · λ ¯m (x) T ¯ y = λ(x) = λ ¯ where y = λ(x) is the measured output, is both feedback linearizable and observable, and (ii) the robust feedback linearized system x˙ r = Ar xr + Br vr y = C¯r xr
(10)
¯ with C¯r = ∂x λ(0), is observable, that is, the pair (C¯r , Ar ) is observable. Proof: (i) System (9) is feedback linearizable since it is assumed that (1) satisfies the conditions for feedback linearization. To analyze the observability of system (9), the observability map, proposed by Zeitz (1984) (see also, e.g., Maggiore and Passino, 2003), is used. Roughly speaking, the observability map is constructed by differentiating each function of the output y until the order equal to its relative degree minus 1. This map is invertible if, and only if, the state x may be written uniquely as a function of y, u and their respective derivatives, which means that the system is observable. ¯ i (x) satisfy For system (9), since the functions λ Lemma 1(a), the observability map does not depend on the input u nor its derivatives and is given by ¯ λi (x) ω1 (x) ¯i (x) Lf λ ω(x) = ... , ωi (x) = (11) .. . ωm (x) ri −1 ¯ L λi (x) f
From Lemma 1(c), ω(x) corresponds to the local diffeomorphism φ(x). Consequently, the observability map is invertible in U (since a diffeomorphism is always invertible). Therefore, system (9) is observable.
(ii) Calculating ∂x ω(0) yields ¯ 1 (0) ∂x λ c¯r1 ¯ 1 (0)∂x f (0) c¯r1 Ar ∂x λ .. .. . . ¯1 (0) (∂x f (0))r1 −1 c¯r Ar1 −1 ∂x λ 1 r .. .. ∂x ω(0) = = . . ¯ ∂x λm (0) c¯rm ¯ ∂x λm (0)∂x f (0) c¯rm Ar .. .. . . rm −1 rm −1 ¯ c¯rm Ar ∂x λm (0) (∂x f (0)) (12) where c¯r1 , . . . , c¯rm are the rows of matrix C¯r . Since ω(x) corresponds to the local diffeomorphism φ(x), it is known that ω −1 (x) exists and both ω(x) and ω −1 (x) are differentiable. By the inverse function theorem, ∂x ω(x) is nonsingular and, in particular, that ∂x ω(0) is nonsingular. Hence, ∂x ω(0) corresponds to the n linearly independent rows of the observability matrix of the linearized system (10), which proves that the observability matrix has rank n. Therefore, the pair (C¯r , Ar ) is observable.
4. ROBUST FEEDBACK LINEARIZATION WITHOUT FULL STATE INFORMATION When all state variables of a nonlinear system are known, it is possible to use the robust feedback linearization given in Lemma 2 to linearize that system. When the state is not known, it is clearly impossible to do it. But, if the nonlinear system satisfies the conditions in Theorem 1, then one can use an observer to obtain an estimation of the state and replace this estimation in the control feedback law (7). If a linear observer is used, then an estimation of the state x ˆr of the linearized system is obtained. The estimation of the state x ˆ of the original nonlinear system is calculated using the “inverse” diffeomorphism xˆ = φ−1 xr ). r (ˆ This modified feedback linearizing loop is shown in Figure 1. vr-
ur-
αr (ˆ x) + βr (ˆ x)vr
Nonlinear System
y-
6 x ˆ
Linear Observer φ−1 xr ) r (ˆ
x ˆr
Fig. 1. Feedback linearizing loop with observer. As said in the introduction, it is not clear whether the linear observer works or not, due to the possible vicious circle mentioned there. The next theorem proves that the robust feedback linearization
fˆ(xr , vr , ǫr ) = θr ◦ σr f ◦ σr + θr ◦ σr g ◦ σr αr ◦ σ ˆr ◦ η + θr ◦ σr g ◦ σr βr ◦ σ ˆr ◦ η vr fˆ(xr , vr , ǫr ) = fˆ(0, 0, 0) + ∂xr fˆ(0, 0, 0)xr + ∂vr fˆ(0, 0, 0)vr + ∂ǫr fˆ(0, 0, 0)ǫr + O(x2r , vr2 , ǫ2r )
makes it possible to use such observer, since the estimation error converges to zero. In other words, it is proved that a separation principle holds. Theorem 2. Consider the nonlinear system x˙ = f (x) + g(x)u ¯ y = λ(x)
(15)
¯ 0 ) = 0. If the nonlinear system (15) with y0 = λ(x satisfies the conditions in Theorem 1 and vr is a stabilizing control for the linearized system (10), then x ˆ˙ r = Ar x ˆr + Br vr + Lobs (y − C¯r xˆr )
(16)
combined with the diffeomorphism xˆ = φ−1 xr ) , σ ˆr (ˆ xr ) r (ˆ
(17)
is a locally exponentially stable observer for system (15) controlled by the robust linearizing feedback ur (ˆ x, vr ) = αr (ˆ x) + βr (ˆ x)vr
(18)
where αr (ˆ x), βr (ˆ x) and φr (ˆ x) are defined according to Lemma 2. Furthermore, the estimation error defined as ǫr = xr − x ˆr
(19)
converges locally exponentially to zero. Proof: Applying the control law u = ur (ˆ x, vr ) to system (15) yields x˙ = f (x) + g(x) αr (ˆ x) + βr (ˆ x)vr (20) Since xr = φr (x), it follows that
x˙ r = ∂x φr (x)x˙ , θ(x)x˙
(21)
Replacing x˙ from (20) in this expression yields x˙ r = θ(x) f (x) + g(x) αr (ˆ x) + βr (ˆ x)vr (22) The state x may be written as x=
φ−1 r (xr )
, σr (xr )
(23)
while the estimated state x ˆ is given by (17). From (19), x ˆr = xr − ǫr , η(xr , ǫr )
(24)
Replacing (17), (23) and (24) in (22), it is possible to rewrite that equation in terms of xr , vr and ǫr , x˙ r = fˆ(xr , vr , ǫr )
(25)
with fˆ(xr , vr , ǫr ) given in (13) and fˆ(0, 0, 0) = 0.
(13) (14)
The function in (13) may be also written as in (14), where O(x2r , vr2 , ǫ2r ) represents the high-order terms of the Taylor expansion about (0, 0, 0). In order to calculate the partial derivatives of fˆ(xr , vr , ǫr ) with respect to xr , vr and ǫr about the fixed point (0,0,0), the following special properties of the functions η, σr , σ ˆr , θr , f , g, αr and βr are used: f (0) = 0 g(0) = Br αr (0) = 0 βr (0) = Im σr (0) = 0 σ ˆr (0) = 0 θr (0) = In
(26) (27) (28) (29) (30) (31) (32)
η(0, 0) = 0 ∂xr σr (0) = In ∂xˆr σ ˆr (0) = In ∂x f (0) = Ar ∂x αr (0) = 0 ∂xr η(0, 0) = In ∂ǫr η(0, 0) = −In
(33) (34) (35) (36) (37) (38) (39)
Using the chain rule to calculate the derivatives of composed functions and applying properties (26)(39), one obtains after some algebra ∂ fˆ ∂f ∂σr (0, 0, 0) = (0) (0) ∂xr ∂x ∂xr (40) ∂αr ∂σ ˆr ∂η + Br (0) (0) (0, 0) = Ar ∂x ∂x ˆr ∂xr ∂ fˆ (0, 0, 0) = Br (41) ∂vr ∂ fˆ ∂αr ∂σ ˆr ∂η (0, 0, 0) = Br (0) (0) (0, 0) = 0 (42) ∂ǫr ∂x ∂x ˆr ∂ǫr Replacing (40), (41) and (42) in (14) and neglecting the high-order terms O(x2r , vr2 , ǫ2r ) yields fˆ(xr , vr , ǫr ) = Ar xr + Br vr
(43)
and it follows from (25) that x˙ r = Ar xr + Br vr The estimation error dynamics is given by ǫ˙r = x˙ r − x ˆ˙ r
(44)
(45)
Using (16) and (44), the estimation error dynamics (45) may be rewritten as ǫ˙r = (Ar − Lobs C¯r )ǫr (46) ¯ From Theorem 1(ii), the pair (Cr , Ar ) is observable. Therefore, it is possible to choose a gain Lobs such that all eigenvalues of (Ar − Lobs C¯r ) lie in the left-half of the complex plane. Hence, the estimation error (19) converges locally exponentially to zero and the linear observer (16) combined to the diffeomorphism (17) is a locally exponentially stable observer for the nonlinear system (15) controlled by the robust linearizing feedback (18).
Remark 1. The statement of the above theorem is not valid for the classical feedback linearization. If an analogous procedure is made with the Brunovsky canonical form, it is not possible to ensure that the estimation error converges to zero. Properties (26)-(39) play a crucial role for this convergence, especially properties (28), (29), (32) and (37), which characterize the robust feedback linearization as seen in Lemma 2.
which yields a good performance/robustness tradeoff, as seen in the simulations. The estimation of the state x ˆ is obtained from the diffeomorphism x ˆr1 x ˆr2 x ˆr3 x ˆ = r 8I 2 x ˆ 2I (ˆ x −ˆ x ) (ˆ x +I )2 0
(k+2ˆ xr1 )
r4
k2
r3
−
0
k3
r1
+
r3
0
(k−2ˆ xr1 )2
−I0
(50)
5. APPLICATION EXAMPLE Consider the magnetic bearing system described by Queiroz and Dawson (1996) (see also Franco et al., 2006). Defining x1 as the rotor position, x2 as the rotor velocity, x3 and x4 as the currents, T u = u1 u2 as the input voltages, this system may be put in the state-space form x2 0 0 L0 (x3 +I0 )2 − (x4 +I0 )2 m (k−2x1 )2 (k+2x1 )2 0 0 u x˙ = R1 (k−2x1 )x3 2x2 (x3 +I0 ) + k−2x1 − 0 − L L k−2x 0 0 1 k+2x R2 (k+2x1 )x4 2x2 (x4 +I0 ) 1 0 − + k+2x1 L0 L0 T y = x1 x3 (47) where m is the rotor mass, I0 is the premagnetization current, R1 and R2 are the resistances of the circuits and L0 and k are positive constants depending on the system construction. Their nominal values are m = 2 kg, k = 2.0125 × 10−3 m, L0 = 3 × 10−4 Hm, R1 = R2 = 1 Ω and I0 = 2 × 10−2 A. The equilibrium point is x0 = 0. System (47) satisfies the conditions of Theorem 1, thus it is feedback linearizable and observable. The robust (47) is 0 8L0 I02 mk3 x˙ r = 0 0 10 0 y= 00 1
feedback linearized form of system 1 0 −2I0 k 2I0 k
0 x 0 r
0
0
2L0 I0 −2L0 I0 mk2 mk2 xr −kR1 0 L0 −kR2 0 L0
0 0 0 0 + Lk 0 vr 0 0 Lk0
(48)
By Theorem 1, system (48) is observable. Observer: A linear observer in the form (16) is designed for system (48). Since the zeros of this system are at infinity, rapid eigenvalues can be chosen for the matrix (Ar − Lobs C¯r ). For the set of eigenvalues {−36, −40, −44, −32}, the observer gain matrix obtained with the robust pole assignment algorithm from Matlab place (Kautsky et al., 1985) is T 44.0 370.6 −91.1 −2736.6 Lobs = (49) −3.1 −156.2 94.5 719.6
Robust Feedback Linearization: According to Lemma 2, αr (ˆ x) and βr (ˆ x) are calculated using α(ˆ x), β(ˆ x), φ(ˆ x), L, T and R given below. ˆ2 (ˆ x3 +I0 ) 0x R1 xˆ3 + 2L(k−2ˆ 2 x1 ) α(ˆ x) = (51) 2 2ˆ x2 L0 (k+2ˆ x1 )(ˆ 3 +I0 ) R2 x ˆ4 + (k−2ˆx1 )3 (ˆx4x+I ) 0 β(ˆ x) =
"
L0 k−2ˆ x1 −m(k+2ˆ x1 ) L0 (k+2ˆ x1 )(ˆ x3 +I0 ) 2(ˆ x4 +I0 ) (k−2ˆ x1 )2 (ˆ x4 +I0 )
0
#
x ˆ1 x ˆ2
φ(ˆ x) = L0 (ˆx3 +I0 )2 (ˆ x4 +I0 )2 m (k−2ˆx1 )2 − (k+2ˆx1 )2 x ˆ3 " # 0 R1 2I0 R2 0 0 −2I mk mk L= −kR1 0 0 −2I 0 k L0 1 0 0 0 0 1 0 0 T = 8L0 I02 2L0 I0 −2L0 I0 mk3 0 mk2 2 mk 0 0 1 0 " # L0 0 R = −mk Lk0 2I0
(52)
(53)
(54)
(55)
(56)
k
H∞ Controller: Since the state of system (48) is estimated, a controller vr = Kr x ˆr designed for the system with transfer matrix Gr (s) = (sI − Ar )−1 Br may be used. The H∞ method with loopshaping proposed by McFarlane and Glover (1992) allows to calculate the controller Kr analytically. For the loop-shaping, the weighting matrices W1 and W2 are chosen such that the frequency response of W2 Gr W1 (and consequently, the frequency response of Kr Gr ) is appropriate. Here are chosen W1 = I and 500(s + 2) W2 = diag , 95, 35, 15 (57) s The weighting matrix W2 adds an integrator to the first row of W2 Gr W1 , which is related to the rotor position x1 , to avoid steady-state errors, and a zero to better shape the position response. To the other lines of W2 Gr W1 , related to the velocity x2 and the currents x3 and x4 , only gains are added. A value γr ≈ 4 is used to calculate the controller. This yields a robustness index of 25%.
Simulations: A comparison with the linear controller Kr (combined with the linear observer) directly applied to the nonlinear system is presented below, to illustrate the performance of the closed loop with robust feedback linearization without full state information. Parameter variations are included, to evaluate the robustness of the method as well. The simulations are carried out with Simulink/Matlab, using the Dormand-Prince algorithm with a maximal step size of 0.1 ms. For these simulations, it is supposed that the parameter variations may be ±10% for m, L0 , R1 and R2 and ±5% for k, yielding 32 different combinations of the extreme values, which are all tested. The initial condition for the position x1 is 0.55mm, which allows the evaluation of the behavior of the controllers far from the equilibrium point. The results for (i) the closed loop with the linear control directly applied to the nonlinear system and (ii) the closed loop with the robust feedback linearization without full state information are given in Fig. 2 and 3 respectively. Linear Control Nominal Parameter Variation
Position x1 (mm)
0.8 0.6 0.4 0.2 0 −0.2 −0.4
0
0.5
1
1.5
2 Time (s)
2.5
3
3.5
4
Fig. 2. Rotor position x1 for (i). Robust Feedback Linearization without Full State Information Nominal Parameter Variation
Position x1 (mm)
0.8 0.6 0.4 0.2 0 −0.2 −0.4
0
0.5
1
1.5
2 Time (s)
2.5
3
3.5
4
Fig. 3. Rotor position x1 for (ii). The closed loop (ii) behaves as desired, having a good nominal performance, and for all the considered parameter variations the response remains close to the nominal one. The closed loop (i) presents a bad nominal performance and is unstable for 5 combinations of the parameter variations.
6. CONCLUDING REMARKS As shown by the theory in Section 4 and illustrated by the simulations in Section 5, it is possible to
perform the robust feedback linearization without full system information of a nonlinear system whose state is unknown. In particular, Theorem 2 proves that the state of the nonlinear system can be reconstructed with a linear observer and a diffeomorphism (provided that robust feedback linearization is used), which is a kind of separation principle. The same does not hold true when classical feedback linearization is used, as explained in Remark 1. Results connecting the concepts of feedback linearizable system and observable system are given in Section 3. In future works, the robustness of the approach (indicated by the simulations results) and the optimal choice of the observer gain matrix will be further studied.
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