INT. J. CONTROL,
2003, VOL. 76, NO. 6, 537–556
Exact feedforward linearization based on differential flatness VEIT HAGENMEYER{* and EMMANUEL DELALEAU{ This article deals with the trajectory aspect of differential flatness from a feedforward point of view. The notion of exact feedforward linearization based on differential flatness is introduced: a differentially flat system, to which a nominal feedforward deduced from flatness is applied, is equivalent, by change of coordinates, to a linear multivariable Brunovsky´ form if the initial condition is consistent with the one considered in the design of the nominal trajectory. In its second part, the new notion states that there exist unique solutions in the vicinity of the desired trajectory when applying a nominal feedforward to the corresponding flat system. To the end of stabilizing the desired trajectory, the information from the Brunovsky´ form is used to design the combination of the nominal feedforward and an additional feedback part. In the case of extended PID controls for the latter, stability is proven using a theorem by Kelemen. Thus the overall control structure turns out to be quite simple and effective for industrial application. Simulations of a DC drive example and experimental results of a magnetic levitation system illustrate its performance.
1.
Introduction
Differential flatness has been presented in both the differential algebraic setting (Fliess et al. 1995) and the differential geometric setting of infinite jets and prolongations (Fliess et al. 1999). In both cases motion planning issues have been addressed and flatness has been shown to be a property which is related to the trajectories of a system. Moreover, it has been established that every flat system is linearizable by endogenous feedback (a special class of dynamic feedback). Furthermore, flatness has allowed to develop the control of many practical examples and led, in several cases, to industrial applications (see, e.g. Le´vine 1999 or Jadot et al. 2000). Even though exact feedback linearization (Jakubczyk and Respondek 1980, Nijmeijer and van der Schaft 1990, Isidori 1995) is an important and a well-known subject, considering flatness only in this context is too restrictive. Hence, taking this argument as a motivation, the notion of exact feedforward linearization based on differential flatness is introduced in this article in order to emphasize, in a new way, that the property of differential flatness can also be considered to design control laws which do not exactly feedback linearize the non-linear system.
Received 10 January 2002. Revised 20 December 2002. * Author for correspondence. e-mail: Veit.Hagenmeyer@ lss.supelec.fr { Laboratoire des signaux et syste`mes, CNRS-Supe´lecUniversite´ Paris-sud, Plateau de Moulon, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette, France. { Department of Electrical and Computer Engineering, Northeastern University, 360 Huntington Avenue, Boston, MA 02115, USA. e-mail:
[email protected]. Also with Universite´ Paris-sud, 15 avenue G. Cle´menceau, 91405 Orsay, France. e-mail:
[email protected]; on leave from Laboratoires des signaux et syste`mes.
The contribution of the article is fourfold. Firstly, it shows that a differentially flat system, to which a nominal feedforward deduced from its flatness is applied, is equivalent, by change of coordinates, to a linear multivariable Brunovsky´ form (Brunovsky´ 1970) without closing the loop if the initial condition is consistent with the one considered in the design of the nominal trajectory. Note that this is a quite new point of view of the property of differential flatness which was previously shown to be a possibility to obtain a Brunovsky´ form using dynamic endogenous feedback (Fliess et al. 1995, 1999). Secondly, it demonstrates that if the initial condition, which is taken into consideration for the design of the nominal feedforward, is not equal but close to the right initial condition, then a unique solution exists for the non-linear flat system in the vicinity of the desired solution of the aforementioned Brunovsky´ form. This also relates flatness to the existence of solutions of non-autonomous systems of differential equations. Thirdly, the article introduces a new tool to study the stability of any closed-loop strategy of flatness-based control schemes. In the context of exact feedforward linearization, one of the different possibilities of feedback control is necessary in order to stabilize deviations from the desired trajectory. Therefore the overall control law consists of two parts: the feedforward signal based on differential flatness, which steers the system when being on the desired trajectory, and a feedback control part, which forces the system to converge to the desired trajectory. The way in which these two parts should be combined is studied by taking the structure of the flat systems into account: via exact feedforward linearization based on differential flatness it is possible to indicate the point in the formula of the nominal feedforward signal at which the outputs of a feedback control should be added in view of stabilizing deviations from the desired trajectory. Stability of this control scheme can be demonstrated when using simple
International Journal of Control ISSN 0020–7179 print/ISSN 1366–5820 online # 2003 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/0020717031000089570
538
V. Hagenmeyer and E. Delaleau
extended PID controls for the feedback part. By considering a stability result by Kelemen (1986), it can be shown that the absolute values of the control coefficients have to be traded off with the velocity of the desired trajectory. Thus, given reasonable bounded initial errors, non-linear flat systems can be stabilized around given desired trajectories by applying exact feedforward linearization and extended PID control. Fourthly, the article illustrates the practicability for real applications of this control scheme. Note that the presented control laws are very simple to be calculated on a micro-controller in real time due to the extended PID feedback part. In addition, they respect the nonlinear nature of the system by applying the flatness based feedforward part, which can either be precalculated or calculated with a reasonable small sample frequency. Moreover, since feedforward linearization based on differential flatness does not necessarily require full state information in many practical examples—in contrast to exact feedback linearization—some sensors or a possible observer may become obsolete in these cases. Furthermore, feedforward linearization does not imply the same well-known robustness problems of exact feedback linearization, which allows for a far larger domain of application. The article is organized as follows: The next section (} 2) introduces the subject via some simple examples, showing all relevant notions related to exact feedforward linearization based on differential flatness; in particular, the robustness issue is described in } 2.3. Section 3 presents the notion of exact feedforward linearization based on differential flatness. Using this result, } 4 establishes a specific control design methodology which leads to a specific error dynamics discussed in } 5. The control design is exemplified by extended PID control which is presented in } 6. In } 7 Kelemen’s result is recalled as elaborated by Lawrence and Rugh (1990) before stability for the proposed control strategy is shown. This strategy is applied to a separately excited DC drive example in } 8. Experimental results of a magnetic levitation system are presented in } 9 to show how an existing PID control can be improved by exact feedforward linearization. The article is concluded by a discussion of the results in } 10.
x_ ¼ x x3 þ u
ð1Þ
with the state x taking values in R and the scalar input u. The initial condition is denoted by xð0Þ ¼ xo . Since this dynamics is flat with flat output z ¼ x, the input u can be chosen as 3 u ¼ x þ x þ v ð2Þ where x is the desired profile of x. In a first step, let the new input simply be v ¼ x_ . This flatness based open loop control is called exact feedforward linearization, since it yields a Brunovsky´ form for all times if the initial conditions of the system and of the desired trajectory are consistent. This can be seen by coupling (1) and (2) which leads to 3 x_ ¼ x x3 þ x þ x þ v
ð3Þ
The consistency of the initial conditions of the system and of the desired trajectory is defined by x ð0Þ ¼ xo . In this case, equation (3) can be written for t ¼ 0 as x_ ¼ v. This Brunovsky´ form also holds for all times when considering v ¼ x_ (cf. } 3 for details). For the sake of comparison, the feedback linearizing control law (see Nijmeijer and van der Schaft 1990, Isidori 1995) can be denoted as u ¼ x þ x3 þ v
ð4Þ
Note, that the difference between the two methods consists in the fact that where in the control law of exact feedforward linearization (2) the desired trajectory of 3 the state ðx þ x Þ is involved, the control law of exact feedback linearization (4) involves the state ðx þ x3 Þ itself. Thus it is remarked that in contrast to exact feedback linearization the ‘well-behaving’{ term ðx x3 Þ is not cancelled by exact feedforward linearization. The second important difference of the two methods is the difference of the linearity of the respective error dynamics. Exact feedback linearization can lead to a linear error dynamics, which is attained when a linear controller placing the poles is developed after the cancellation of the non-linear terms (Isidori 1995). Applying (4) to (1) leads to x_ ¼ v. Considering thereafter v ¼ x_ þ 1 e (where the tracking error is defined as e ¼ x x ) yields the linear error dynamics e_ ¼ 1 e
2.
Introductory examples
To give a first glance of exact feedforward linearization, simple academic examples are developed before the formal presentation of the general case. 2.1. A SISO example Consider the simple SISO academic example borrowed from Martin et al. (2000)
For exact feedforward linearization, a linear error dynamics cannot be attained. This need not be a disadvantage as can be seen by the following (cf. also the robustness point of view presented in } 2.3). The nonlinear error system for exact feedforward linearization can be found by considering (3) and v ¼ x_ { ‘Well-behaving’ in the sense of convergence behaviour and counter-action with respect to any kind of perturbations.
539
Exact feedforward linearization 2 e_ ¼ eðe2 þ 3x e þ 3x þ 1Þ
This non-linear error system is globally asymptotically 2 stable since ðe2 þ 3x e þ 3x þ 1Þ > 0 for every admis sible x , which can easily be seen by considering e2 þ 3x e þ 3x2 þ 1 ¼ ðe þ 32 x Þ2 þ 34 x2 þ 1. Since for the stability of the general case presented in } 7 the linearization of the error system around the origin is studied, it is calculated here for the example 2 as e_ ¼ ð3x þ 1Þe . Note that the time dependency of the coefficient of this linearization does not interfere with the global asymptotic convergence behaviour of the system (for the general case, see } 7 for details). The convergence behaviour can be modified when considering the same new input v ¼ x_ þ 1 e as in the exact feedback linearization case. This leads to 2 e_ ¼ eðe2 þ 3x e þ 3x þ 1Þ þ 1 e
Note that the linearization of this system around the 2 origin is e_ ¼ ð3x þ 1 1 Þe . Thus when considering a time dependent coefficient 1 ðtÞ, it is possible to place the pole of the local error system. In the general context of exact feedforward linearization, the new input is chosen to be v ¼ x_ þ LðeÞ, where LðeÞ represents a degree of freedom with respect to the design of one of the different possibilities of stabilizing feedback (sliding modes, Lyapunov-based, backstepping, etc.). In this article, simple PID-like controls are considered to counteract deviations from the desired trajectory. Hence the new input reads for the example as ðt v ¼ x_ þ 0 eðÞ d þ 1 eðtÞ
1;1 ¼ x1 2;1 ¼ x2 2;2 ¼ x3 þ x2 u1
9 > > = > > ;
ð6Þ
Note that in the MIMO case the state transformation may depend on the input. Then (5) reads as 9 > _1;1 ¼ u1 > > = ð7Þ _2;1 ¼ 2;2 > > > _2;2 ¼ 2;2 u1 þ 2;1 u_ 1 þ u2 ; In Delaleau and Rudolph (1998) it was shown that the following set of algebraic equations taken from (7) ) u1 ¼ v 1 ð8Þ 2;2 u1 þ 2;1 u_ 1 þ u2 ¼ v2 always yields a solution for u; it is in this case ) u1 ¼ v 1 u2 ¼ v2 2;2 v1 2;1 v_1
ð9Þ
Considering (7) and (8) it can be seen that v1 and v2 can be replaced by _1;1 and _2;2 respectively for the algebraic solution of (8), thus the exact feedforward linearization control law can be deduced from (9) directly. Therefore the nominal feedforward of u reads as ) u1 ¼ _1;1 ð10Þ _ € u2 ¼ _2;2 2;2 1;1 2;1 1;1
2.2. A MIMO example
Under the consistency of the initial condition, that is (cf. (6)) 9 ð0Þ ¼ x1 ð0Þ 1;1 > > = 2;1 ð0Þ ¼ x2 ð0Þ ð11Þ > > ; 2;2 ð0Þ ¼ x3 ð0Þ þ x2 ð0Þð_1;1 ð0Þ þ 1;1 ð0ÞÞ
In the following, a MIMO academic example is presented to highlight the concept of exact feedforward linearization in the multi-variable context in the first approach. Given the MIMO system{ (cf. Isidori 1995, p. 302) 9 x_ 1 ¼ u1 > > = ð5Þ x_ 2 ¼ x3 þ x2 u1 > > ; x_ 3 ¼ u2
the application of the nominal feedforward (10) to (7) is equivalent to the following linear system in multivariable Brunovsky´ form for all times (see Proposition 1 in } 3 for details) 9 _ 1;1 ¼ _1;1 > > > = _ ¼ ð12Þ 2;2 > 2;1 > > ; _ 2;2 ¼ _2;2
0
where xð0Þ ¼ xo . This system is flat with flat output z ¼ ½x1 ; x2 T . It can be transformed into the so-called Brunovsky´ state via the coordinate change { Note that this example is not static state linearizable (Jakubczyk and Respondek 1980).
Thereby _1;1 and _2;2 play the role of the inputs to the respective subsystem. In the second part of Proposition 1 in } 3 it is established that unique solutions exist at least for a finite time interval in the vicinity of the desired trajectory when the initial condition is non-consistent, but not ‘too far away’ from the consistent one. These solutions
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V. Hagenmeyer and E. Delaleau
are stabilized using an extended PID control structure{ as described in } 6 9 > u1 ¼ _1;1 þ 1;1;1 e1;1 > > > > > 2 = X _ _ u2 ¼ 2;2 þ 2;2; j e2; j 2;2 1;1 ð13Þ > > j¼1 > > > > ; € 2;1 1;1 where the error e ¼ n n . Considering the application of this control (13) to the system (7), the tracking error dynamics can be established as in } 2.1. For the MIMO example it reads as 9 e_1;1 ¼ 1;1;1 e1;1 > > > > > > > e_2;1 ¼ e2;2 > > = 2 € ð14Þ e_2;2 ¼ ð2;2;1 þ 1;1 þ 1;1;1 e1;1 Þe2;1 > > > > þ ð2;2;2 þ _1;1 þ 1;1;1 e1;1 Þe2;2 > > > > > ; þ ð2;1 1;1;1 þ 2;2 Þ1;1;1 e1;1 Its stability can be determined by studying its linearization around the origin: this fact is established by Proposition 2 in } 7. The linearized system around the origin is given by 2 3 1;1;1 0 0 6 7 0 1 7 e_ ¼ 6 ð15Þ 4 0 5 e 2;1;1 2;2;1 2;2;2 where 9 > > > > = ¼ ð2;1 1;1;1 þ 2;2 Þ1;1;1 >
1;1;1 ¼ 1;1;1 2;1;1
2;2;1 ¼ 2;2;1 þ €1;1 2;2;2 ¼ 2;2;2 þ _1;1
> > > > > ;
ð16Þ
The corresponding characteristic polynomial reads ðs 1;1;1 Þðs2 ð2;2;2 þ _1;1 Þs ð2;2;1 þ €1;1 ÞÞ
ð17Þ
To obtain negative eigenvalues, the coefficients have to be chosen such that for all times 1;1;1 < 0, 2;2;1 < _1;1 and 2;2;2 < €1;1 . If the absolute values of 1;1;1 and 2;2;2 are chosen to be not ‘too big’, then for all t for which _1;1 < 0 and €1;1 < 0, the absolute values of _1;1 and €1;1 have to be balanced carefully with the absolute values of 2;2;1 and 2;2;2 respectively. Then Proposition 2 in } 7 guarantees stability. { The integrals are omitted in the MIMO example for the sake of simplicity.
2.3. Robustness issue After having presented a MIMO academic example, the SISO example of } 2.1 is pursued in order to clarify in a first approach that exact feedforward linearization does not imply the same well-known robustness problems of exact feedback linearization. To this end, the real system of the model (1) is considered for instance to be x_ ¼ 0:9x 0:9x3 þ u
ð18Þ
In contrast to exact feedback linearization, in the case of exact feedforward linearization there does not exist the possibility of an imperfect cancellation of nonlinearities, which may be the origin of a destabilization of the system. In the case of exact feedback linearization, this imperfect cancellation of non-linearities may lead to the fact that the controller induces involuntarily ‘badly’ behaving terms. Substituting (4) into the perturbed system (18) leads to x_ ¼ 0:1x þ 0:1x3 þ v
ð19Þ
The new input v has to counteract in this case the ‘instability term’ ð0:1x þ 0:1x3 Þ introduced by the feedback linearizing control law (4). Considering as above v ¼ x_ þ 1 e yields the following non-linear error equation 2 e_ ¼ e 0:1ðe2 þ 3x e þ 3x þ 1 þ 101 Þ 3
þ 0:1ðx þ x Þ To have asymptotic stability for the ‘inner’ system e_ ¼ e 0:1ðe2 þ 3x e þ 3x2 þ 1 þ 101 Þ, the following 2 condition must hold 1 < 0:1ðe2 þ 3x e þ 3x þ 1Þ. 2 2 Since the term ðe þ 3x Þ might rapidly become large depending on the control task, it can be seen at first glance, where the well-known robustness problems of exact feedback linearization stem from. In the case of exact feedforward linearization, the ‘well-behaving’ term ð0:9x 0:9x3 Þ is not cancelled, therefore its passivity property is used for the stability (even without using any feedback!). Coupling (2) and (18) results in 3 x_ ¼ 0:9x 0:9x3 þ x þ x þ v
ð20Þ
Considering v ¼ x_ leads to the non-linear error system for exact feedforward linearization 2 3 e_ ¼ e 0:9ðe2 þ 3x e þ 3x þ 1Þ þ 0:1ðx þ x Þ
Reasoning as in } 2.1 the ‘inner’ system e_ ¼ e 0:9ðe2 þ 3x e þ 3x2 þ 1Þ is asymptotically stable; thus the whole error system is L2 -stable with respect to the 3 so-called ‘quasi-exogenous’ perturbation 0:1ðx þ x Þ. To modify the convergence behaviour and to obtain a
541
Exact feedforward linearization vanishing tracking error, a PID-like control is used in the feedback part (as already described in } 2.1) ðt v ¼ x_ þ 0 eðÞ d þ 1 eðtÞ 0
3.
Exact feedforward linearization based on differential flatness Differential flatness is a structural property of a class of multi-variable non-linear systems, for which, roughly speaking, all system variables can be written in terms of a set of specific variables (the so-called flat outputs) and their derivatives. Given the multi-variable{ non-linear system x_ ðtÞ ¼ fðxðtÞ; uðtÞÞ;
xð0Þ ¼ x0
ð21Þ
with time t 2 R, state xðtÞ 2 Rn and input uðtÞ 2 Rm . The vector field f : Rn Rm ! TRn is smooth. The system (21) is said to be (differentially) flat (Fliess et al. 1995, 1999) if there exists a set of m differentially independent variables z ¼ ½z1 ; . . . ; zm T such that z ¼ Aðx; u; u_ ; . . . ; uð Þ Þ
ð22Þ
x ¼ /ðz; z_ ; . . . ; zð Þ Þ
ð23Þ
u ¼ wðz; z_ ; . . . ; zð þ1Þ Þ
ð24Þ
where A, / and w are smooth functions of their arguments at least in an open{ subset of Rnþmð þ1Þ , Rmð þ1Þ and Rmð þ2Þ respectively. Such a z is called a flat output. The equations (23) and (24) yield, that for every given trajectory of the flat output t 7! zðtÞ, the evolution of all other variables of the system t 7! xðtÞ and t 7! uðtÞ is also determined without integration of the system of differential equations. Moreover, given a sufficiently smooth desired trajectory—called the nominal trajectory—for the flat output t 7! z ðtÞ, equation (24) can be used to design the corresponding feedforward u directly— called the nominal control. The family of nominal feedforwards is given by u ¼ wðz ; z_ ; . . . ; z
ð þ1Þ
Þ
ð25Þ
that is, for each admissible nominal trajectory t 7! z ðtÞ, there is a nominal feedforward t 7! u ðtÞ. Definition 1: The desired trajectory of the flat output t 7! z ðtÞ is consistent with the initial condition x0 , if (cf. (23)) x0 ¼ /ðz ð0Þ; z_ ð0Þ; . . . ; z
ð Þ
ð0ÞÞ
{ A SISO version of this work is presented in Hagenmeyer & Delaleau (2002). { Note that only a finite number of isolated singularities can arise for control systems of practical interest.
Exact feedforward linearization based on differential flatness is established by the following proposition. Proposition 1: If the desired trajectory of the flat output t 7! z ðtÞ is consistent with the initial condition x0 , then, when applying the nominal feedforward ð25Þ to the differentially flat system given by ð21Þ, the latter is equivalent, by change of coordinates, to a linear system in multi-variable Brunovsky´ form with m chains of integrators for all times (see ð40Þ below). Moreover, if the desired trajectory of the flat output z ðtÞ is not consistent with the initial condition x0 , but x0 is sufficiently close to the initial condition defined by /ðz ð0Þ; z_ ð0Þ; . . . ; z ð Þ ð0ÞÞ, then, when applying ð25Þ to ð21Þ, there exists a unique solution of ð21Þ at least for a given finite time interval in the vicinity of the desired trajectory, which represents the solution of the aforementioned system in multi-variable Brunovsky´ form. Proof: Considering the results of Rudolph (1995) and Delaleau and Rudolph (1998), every flat system can be represented using a so-called Brunovsky´ state (see Appendix A for its construction).{ Setting{ 9 ð 1Þ > n ¼ ½z1 ; z_ 1 ; . . . ; z1 1 ; z2 ; . . . ; > > > > > ð m1 1Þ ð m 1Þ T = zm1 ; zm ; z_m ; . . . ; zm ð26Þ > ¼ ½1;1 ; 1;2 ; . . . ; 1; 1 ; 2;1 ; . . . ; > > > > > T; m1; m1 ; m;1 ; m;2 ; . . . ; m; m P where m i¼1 i ¼ n, the system (21) can be transformed via a well-defined state transformation (Fliess 1990, Fliess et al. 1994) between x and n (parametrized by u and its derivatives) n ¼ Bðx; u; u_ ; . . . ; uð%Þ Þ
ð27Þ
into the normal form _i; j ¼ i; jþ1 ;
j 2 f1; . . . ; i 1g
_i; i ¼ i ðn; u; u_ ; . . . ; uði Þ Þ;
9 =
i 2 f1; . . . ; mg ;
ð28Þ
where i , i 2 f1; . . . ; mg; are also smooth with respect to their arguments and n0 ¼ nð0Þ ¼ Bðx0 ; uð0Þ; u_ ð0Þ; . . . ; uð%Þ ð0ÞÞ
ð29Þ
{ The Brunovsky´ state is chosen here in the following for the sake of preserving the system dimension. Note that it is also possible to choose instead of the transformation into the Brunovsky´ state a transformation into a similar normal form using a classical dynamical extension as in Fliess et al. (1995, 1999). Depending on the system, this can be preferable with respect to the preservation of symmetries or the avoidance of certain singularities. { Thereby the convention is used that if i ¼ 0 then ð 1Þ ðzi ; z_i ; . . . ; zi i Þ ¼ 1.
542
V. Hagenmeyer and E. Delaleau
from (27).{ Delaleau and Rudloph (1998) have furthermore shown for all flat systems that there always exists for the set of algebraic equations (cf. (28)) i ðn; u; u_ ; . . . ; uði Þ Þ ¼ vi ;
i 2 f1; . . . ; mg
ð30Þ
a solution u ¼ ?ðn; v; v_ ; . . . ; vðÞ Þ
ð31Þ
T
where v ¼ ½v1 ; . . . ; vm and ¼ max ði Þ, i 2 f1; . . . ; mg. Comparing (28) and (30) clarifies that ð jÞ
vi
ð jþ1Þ
¼ i; i ;
i 2 f1; . . . ; mg;
j2N
ð32Þ
Hence furthermore comparing (24) and (31) yields wðz; z_ ; . . . ; zð þ1Þ Þ ¼ ?ðn; n; n_ ; . . . ; nðÞ Þ
ð33Þ
T
with n ¼ ½_1; 1 ; . . . ; _m; m and þ 1 ¼ max ð i þ i Þ 1, i 2 f1; . . . ; mg. Considering (30), (31) and (32) leads moreover to _ ; . . . ; ?ði Þ Þ ¼ _i; ; i ðn; ?; ? i
i 2 f1; . . . ; mg
ð34Þ
Now the relationship between (23) and the state transformation (27) can be clarified. From (27), (31) and (32) it can be deduced _ ; . . . ; ?ð%Þ Þ x ¼ B 1 ðn; ?; ?
ð35Þ
where ? as in (33). Thus, comparing with (23), one gets /ðz; z_ ; . . . ; zð Þ Þ ¼ B 1 ðn; ?; ?_ ; . . . ; ?ð%Þ Þ
ð36Þ
After the given preliminaries, the proof proceeds by the following statement: applying the feedforward (25) to the differentially flat system given by (21) is equivalent to the application of u of (25) to (28), which results in 9 = j 2 f1; . . . ; i 1g _i; j ¼ i; jþ1 ; ð37Þ ð Þ _i; i ¼ i ðf; ? ; ?_ ; . . . ; ? i Þ; i 2 f1; . . . ; mg ; where ? ¼ ?ðn ; n ; n_ ; . . . ; nðÞ Þ in view of (31) and (33); moreover ð%Þ f0 ¼ fð0Þ ¼ Bðx0 ; ? ð0Þ; ?_ ð0Þ; . . . ; ? ð0ÞÞ
ð38Þ
considering (29). In the following, the solution t 7! uðt; f0 Þ of the non-autonomous system of differential equations (37) is studied. Since the i , i 2 f1; . . . ; mg, in (37) are smooth, at least local existence and uniqueness of the solution uðt; f0 Þ are given for t 2 ½0; 1 , 1 > 0. To distinguish in the following the two cases of consistent and non-consistent initial conditions, ðt; n0 Þ the corresponding solutions are denoted by t 7! u ~ ðt; f0 Þ respectively: and t 7! u 1: The case is investigated, in which the desired flat output t 7! z ðtÞ is consistent with the initial condition { In (27) % can now be specified as % ¼ max ði 1Þ, i 2 f1; . . . ; mg, cf. (28).
x0 , that is x0 ¼ /ðz ð0Þ; z_ ð0Þ; . . . ; z ð Þ ð0ÞÞ. Considering (36) and (38), this statement is equal{ to f0 ¼ n0 . By local existence and uniqueness of the solution, it is ðt; n0 Þ ¼ n ðtÞ at least for the time interevident that u val t 2 ½0; 1 Þ, which can be verified using (37) and ðt; n0 Þ ¼ n ðtÞ is also the soluconsidering (34). But u tion of the differential equation 9 _i; j ¼ i; jþ1 ; j 2 f1; . . . ; i 1g = ð39Þ ; _i; i ¼ _i; ; i 2 f1; . . . ; mg i with f0 ¼ n0 . In this equation the terms _i; are different i for each choice of z and play the role of the inputs of (39). Consequently (39) is called to be in multi-variable Brunovsky´ form composed of m chains of i integrators with inputs vi ¼ _i; respectively. i To prove that ½0; 1 Þ ¼ ½0; 1Þ, consider the following. From the fact, that i in (37) are smooth, local existence, uniqueness and continuity of the solution can be deduced. Therefore it can be shown, that the subset S of ½0; 1Þ such that (37) is equal to (39) is open in ½0; 1Þ. Moreover it can be deduced, that two solutions, which are different at a given time instant, are also different in an open neighbourhood around this time instant. Thus, it is clear that the subset S of ½0; 1Þ such that (37) is equal to (39) is closed in ½0; 1Þ. Hence S is open and closed in ½0; 1Þ and since ½0; 1Þ is connected (but not compact), it follows that S ¼ ½0; 1Þ.
2: The case of non-consistent initial conditions is defined by f0 6¼ n0 . Since the consistent case solution ðt; n0 Þ of (37) is equal to the designed desired n ðtÞ, it u ðt; n0 Þ belongs to a certain D for all is assured, that u t 2 ½0; t1 , where D Rn is an open connected set and 0 < t1 < 1. Then it can be shown by direct application of the theorem of the continuity of existence of solutions (see, for instance, Theorem 2:6 in the book of Khalil (1996)), that given > 0, there is > 0, such that if jjf0 n0 jj <
ð40Þ
~ ðt; f0 Þ of (37) defined on then there is a unique solution u ~ ð0; f0 Þ ¼ f0 , and u ~ ðt; f0 Þ satisfies ½0; t1 , with u jj~ uðt; f0 Þ n ðtÞjj < ;
8t 2 ½0; t1
ð41Þ
The restriction to the time interval t 2 ½0; t1 is valid for unstable systems. If the system is stable, then the result holds for all times. & In other words, for a given size of a tube around n ðtÞ it can always be defined a neighbourhood of the { Remembering (26), it is remarked, that the n0 are defined by the corresponding design of the desired trajectories of the elements of the flat output.
543
Exact feedforward linearization ξ
, i 2 f1; . . . ; mg play the Since in (39) the terms _i; i role of the inputs to the respective Brunovsky´ form, the new input v 2 Rm is designed as
ε
þ Li ðeÞ ¼ vi þ Li ðeÞ; vi ¼ _i; i
ξ∗
i 2 f1; . . . ; mg ð42Þ
where the tracking error e ¼ ½e1;1 ; e1;2 ; . . . ; e1; 1 ; e2;1 ; . . . ; em;1 ; em;2 ; . . . ; em; m T is defined as
e ¼ n n
ð43Þ
which, in view of (26), is equal to δ
ei;1 ¼ zi zi
and
ð j1Þ
ei; j ¼ zi
zi
ð j1Þ
ð44Þ
For the feedback part of (42) it holds in principle that 0 Figure 1.
t1
t
Exact feedforward linearization based on differential flatness.
initial condition n0 with the size , such that trajectories departing in that neighbourhood will stay in the tube for t 2 ½0; t1 (see figure 1). Since it was established for consistent initial conditions, that the system can be interpreted as being a linear one in multi-variable Brunovsky´ form with m chains of integrators, it can be deduced, that for non-consistent initial conditions being in a neighbourhood of n0 the system trajectories evolve in the neighbourhood of trajectories of a linear system in multi-variable Brunovsky´ form with m chains of integrators. 4.
Exact feedforward linearization and control design Taking into consideration the result of Proposition 1, a control design methodology for the control task of tracking desired trajectories is presented in the following.{ It does not exactly feedback linearize the system and avoids therefore the cancellation of ‘well-behaving’ terms. On the contrary, the presented control design methodology linearizes the system exactly by feedforward when being on the desired trajectory and stabilizes it around the latter. Thus, the control law to be designed consists of two parts,{ a feedforward part (25) and a feedback part taking the tracking error into account. The structure of the combination of both parts is established in the following.
,ð0Þ ¼ 0
ð45Þ
Thereby the Li ðeÞ, i 2 f1; . . . ; mg can be any type of control: be it of sliding mode type (see, e.g. SiraRamı´ rez 2000), based on Lyapunov stability theory (see, e.g. Chelouah et al. 1996), backstepping (see, e.g. Martin et al. 2000), H1 /-analysis of the linearized system around the desired trajectory (see, e.g. Cazaurang 1997) or classical PID (see, e.g. Le´vine 1999). Extended PID control is studied in } 6. Considering the design of (42), the combination of the feedforward part (25) and (42) results in the following control structure (cf. also (31), (32) and (33)) u ¼ ?ðn ; v; v_ ; . . . ; vðÞ Þ
ð46Þ
with v as defined in (42). The advantage of this structure becomes evident in view of (34). One gets when being on the desired trajectory{ _ ; . . . ; ?ði Þ Þ ¼ vi ; i ðn ; ?; ?
i 2 f1; . . . ; mg
ð47Þ
where ? as in (46). This yields
9 _ ; . . . ; ?ði Þ Þ > @i ðn ; ?; ? > ¼ i; j > > > @vj > > > = ði Þ _ @i ðn ; ?; ?; . . . ; ? Þ ¼0 > > ðkÞ > > @vj > > > > ; i; j 2 f1; . . . ; mg; k 2 N
ð48Þ
where i; j represents the Kronecker symbol.
5. Structure of the error dynamics { Throughout the rest of the article, the obtained results are presented in n-coordinates for the sake of simplicity. Remark, that the respective control laws could also be designed in n-coordinates and then be recast into the original x-coordinates by using (23). { In the linear case, refer to the control structure advocated in Fliess and Marquez (2000).
To study the behaviour of the system (21) under the control law (46) in the vicinity of the desired trajectory, the control law (46) is applied to (28), which yields { Being on the desired trajectory means n ¼ n , that is also z ¼ z and correspondingly x ¼ x (cf. (23)).
544
V. Hagenmeyer and E. Delaleau
_i; j ¼ i; jþ1 ;
9 =
j 2 f1; . . . ; i 1g
_ ; . . . ; ?ði Þ Þ; _i; i ¼ i ðn; ?; ?
i 2 f1; . . . ; mg
;
ð49Þ
where ? as in (46). Using (49) and (43), the corresponding tracking error system can be denoted as 9 e_i; j ¼ ei; jþ1 ; j 2 f1; . . . ; i 1g > > > = ði Þ _ _ ð50Þ e_i; i ¼ i ðe þ n ; ?; ?; . . . ; ? Þ i; i ; > > > ; i 2 f1; . . . ; mg The linearized system around the desired trajectory (e ¼ 0) is then given by 3 2 C1;1 C1;2 C1;m 7 6 6 C2;1 C2;2 C2;m 7 7 6 7 6 e_ ¼ 6 ð51Þ 7e .. 7 .. .. 6 .. 6 . . 7 . . 5 4 Cm;1 where each 2 0 6 6 0 6 6 6 . . Ci;i ¼ 6 6 . 6 6 6 0 4
1
0
0
1
..
i;i;1
Cm;m
Cm;2
0
0
i;i;2
i;i;3
0
3
i 2 f1; . . . ; mg
..
.
6. Extended PID control In the following, a PID-like stabilization around the desired trajectory is designed for the feedback part of equation (42). To this end, for the ith-subsystem first a PIDki;i controller{ is developed, which feeds back the respective errors ei; j of the same ith subsystem up to the degree j ¼ ki;i þ 1. Second, if necessary, similar PIDki;l controllers for the ith-subsystem are developed to counteract couplings with the respective lth-subsystem in the vicinity of the desired trajectory. After the following definition ðt ei;0 ¼ ei;1 ðÞ d; i 2 f1; . . . ; mg ð56Þ
Li ðeÞ ¼
0
3
l6¼i
i;l; j el; j ;
j¼0
where ki;i are fixed integers in f1; . . . ; i 1g, ki;l are fixed integers{ in f1; 0; . . . ; l 1g and i;l;j 2 R. Thus, the control structure given in (46) can comprisingly be denoted by u ¼ ?ðn ; v; v_ ; . . . ; vðÞ Þ
7 .. 7 . 7 7 7; 7 0 7 5
vi ¼ _i; þ i
kX i;i þ1
i;i; j ei; j
j¼0
i; j 2 f1; . . . ; mg; j 6¼ i
ð53Þ
is an R i j matrix. Thereby i; j;k ¼ i; j;k þ i; j;k with (because of the arguments of i ðn; u; u_ ; . . . ; uði Þ Þ in (28)) _ ; . . . ; ?ði Þ Þ @j;k @i ðe þ n ; ?; ? i; j;k ¼ @j;k @ej;k e¼0
in view of (43), and furthermore
i;i; j ei; j þ
i;l þ1 X kX
i 2 f1; . . . ; mg ð57Þ
i; j; j
_ ; . . . ; ?ði Þ Þ @i ðn; ?; ? ¼ @j;k
kX i;i þ1 j¼0
ð52Þ
in view of (42) and (48).
the Li ðeÞ in (42) can then be written as
i;i; i
is an R i i matrix and each 2 0 0 6 6 .. 6 . 6 Ci; j ¼ 6 6 0 6 0 4 i; j;1 i; j;2
ð55Þ
0
7 0 7 7 7 .. 7 . 7 7; 7 7 1 7 5
.
i; j;k
_ ; . . . ; ?ði Þ Þ @vi @i ðe þ n ; ?; ? ¼ @vi @ej;k e¼0 @vi @Li ðeÞ ¼ ¼ @ej;k e¼0 @ej;k e¼0
þ
i;l þ1 X kX
l6¼i
j¼0
i;l; j el; j ;
9 > > > > > > > > > = > > > > > > > > i 2 f1; . . . ; mg > ;
ð58Þ
This structure consists of a non-linear combination of a non-linear feedforward part based on differential flatness and a simple linear feedback part of extended PID type (cf. figure 2 in the single input case). Even though the correction term is linear in the tracking error, the overall control scheme is a real non-linear one.
ð54Þ n¼n
{ The notation ‘PIDk controller’ stands for an extended PID controller involving derivative P actions up to the kth order. { Thereby the convention 1 j¼0 i;l; j el; j ¼ 0 is used.
545
Exact feedforward linearization
Trajectory generator
. ξ *,ξ *n
Combination u= ψ(ξ∗, v) . v= ξ n*+ Λ(e)
u
ξ
Nonlinear differentially flat system
Λ( e)
ξ* -
Figure 2.
7.
ξ
Extended PID
Scheme of the combination of exact feedforward linearization and extended PID control in the SISO case.
Stability
In this section it is shown that the proposed control strategy (58) is able to stabilize the system (21) around given desired trajectories z . For the sake of generality, full state information is applied to the PIDki; j parts of the control (58), that is ki; j ¼ i 1, i; j 2 f1; . . . ; mg in (57) in the following. How this assumption can be relaxed thereafter is described at the end of this section (see Remark 4 after the proof of the following proposition for the partial state feedback of PIDki; j parts with 0 ki; j < i 1, i; j 2 f1; . . . ; mg). Nevertheless, using the full state information setting, comparability with the classic feedback linearization approach is given. 7.1. A stability result by Kelemen To study the stability of the proposed methodology, one makes use of a result which was primarily introduced by Kelemen (1986) and reinterpreted by Lawrence and Rugh (1990) and also by Khalil and Kokotovic´ (1991). In the following, the presentation of Lawrence and Rugh (1990) is adopted. Given the system g_ ðtÞ ¼ gðgðtÞ; nðtÞÞ;
gð0Þ ¼ g0 ;
t t0
ð59Þ
where gðtÞ is the n 1 state vector and nðtÞ is the m 1 input vector. We assume that (H1): (H2):
(H3):
g : Rn Rm ! Rn is of class C2 with respect to its arguments, there is a bounded, open set G Rm and a continuously differentiable function ! Rn such that for each constant input n: G value t 2 G, gðnðtÞ; tÞ ¼ 0, there is a > 0 such that for each t 2 G, the eigenvalues of ð@g=@gÞðnðtÞ; tÞ have real parts no greater than .
These hypotheses guarantee that the system (59) has a manifold of exponentially stable constant equilibria, which we have chosen to parameterize by constant
values of the input. In the sequel, we let k k denote the (pointwise in time) Euclidean norm of a (time-varying) vector. Theorem 1 (Kelemen 1986, Lawrence and Rugh 1990): Suppose the system ð59Þ satisfies ðH1Þ, ðH2Þ and ðH3Þ. Then there is a > 0 such that given any 2 ð0; and T > 0, there exists 1 ðÞ; 2 ð; TÞ > 0 for which the following property holds. If a continuously differentiable input t ! nðtÞ satisfies nðtÞ 2 G, T t0 , kg0 nðnðt0 ÞÞk < 1
ð60Þ
and 1 T
ð tþT
k_nðÞk d < 2 ;
t t0
ð61Þ
t
then the corresponding solution of ð59Þ satisfies kgðtÞ nðnðtÞÞk < ;
t t0
ð62Þ
In an important corollary of this theorem, the following properties are established. Corollary 1 (Kelemen 1986, Lawrence and Rugh 1990): If, in addition to the conditions of the theorem, the input signal satisfies lim n_ ðtÞ ¼ 0;
t!1
lim nðtÞ ¼ n1 2 G
t!1
ð63Þ
then the corresponding solution of ð59Þ satisfies lim gðtÞ ¼ nð1 Þ
t!1
ð64Þ
Furthermore, if for some T1 > t0 ; nðtÞ ¼ n1 2 G for all t T1 , then gðT1 Þ is in the domain of attraction of the exponentially stable equilibrium ðnðn1 Þ; n1 Þ. Remark 1: In Kelemen (1986), the theorem required that k_nðtÞk be sufficiently small for each t, while (61) requires that it be sufficiently small in an average sense, uniformly in time.
546
V. Hagenmeyer and E. Delaleau (H1):
7.2. Stability for the proposed control strategy The augmented tracking error system can be found after the definition (remember (56)) e ¼ ½e1;0 ; e1;1 ; . . . ; e1; 1 ; e2;0 ; e2;1 ; . . . ; em;0 ; em;1 ; . . . ; em; m T Then one gets for (50) in view of (58) e_i; j ¼ ei; jþ1 ;
j 2 f0; . . . ; i 1g
_ ; . . . ; ?ði Þ Þ _i; e_i; i ¼ i ðe þ n ; ?; ? i
¼ i ðe; Z Þ;
i 2 f1; . . . ; mg
9 > > = > > ;
ð65Þ
ðÞ
where Z ¼ ½n ; n ; n ; . . . ; n T (cf. (33)) and ¼ max ði Þ, i 2 f1; . . . ; mg (cf. (31)). This system can be written as
_
e_ ¼ Lðe; Z Þ
ð66Þ
where the desired flat output and its þ 1 derivatives involved in Z play the role of an input to the tracking error system in e. Now, in comparing (66) with (59) (that : : is g ¼ e and n ¼ Z ), the stability result of the proposed control strategy can be stated as follows. Proposition 2: There is a > 0 such that for all 2 ð0; and T > 0 there exists 1 ðÞ; 2 ð; TÞ > 0 for which the following property holds. If a sufficiently continuously differentiable desired trajectory z ðtÞ satisfies Z ðtÞ 2 G R þ1 ðG as defined in ðH2Þ of Theorem 1 in } 7.1Þ, T t0 keðt0 Þk < 1
ð67Þ
and 1 T
ð tþT
kZ_ ðÞk d < 2 ;
t t0
ð68Þ
t
then the corresponding solution eðtÞ of ð65Þ satisfies keðtÞk < ;
t t0
ð69Þ
that is the system (21) is stable under the tracking control law (58). Moreover, if in addition the desired trajectory satisfies lim Z_ ðtÞ ¼ 0;
t!1
lim Z ðtÞ ¼ Z1 2 G
t!1
ð70Þ
then the corresponding solution of ð65Þ satisfies lim eðtÞ ¼ 0
t!1
ð71Þ
Furthermore, if for some T1 > t0 ; Z ðtÞ ¼ Z1 2 G for all t T1 , then eðT1 Þ is in the domain of attraction of the exponentially stable equilibrium ð0; Z1 Þ, that is the system (21) is asymptotically stable under the tracking control law (58). Proof: Since Proposition 2 is based on Kelemen’s result as elaborated in Lawrence and Rugh (1990), one proceeds by showing, that the hypotheses (H1)–(H3) of Theorem 1 given in } 7.1 can always be fulfilled:
Straightforward.
(H2): The boundedness of the set G R þ1 can be assured by the fact that only sufficiently smooth bounded desired trajectories z ðtÞ with bounded derivatives are studied. Therefore all point-wise in time inputs R þ1 . That G is closed in the pressatisfy Z 2 G ¼ G ent case and not open as in Kelemen’s theorem does not interfere with its applicability: the hypothesis of openness of G in Kelemen (1986) or Lawrence and Rugh (1990) is due to the hypothesis of existence of a manifold composed of stable equilibria when applying constant inputs from G. In the present case, the manifold of stable equilibria contracts to a single stable equilibrium (for its stability see (H3)): setting the input Z ðtÞ 2 G R þ1 of (65) pointwise in time as constant Z 2 G R þ1 and studying the corresponding equilibria leads to the con ! Rnþm , which is tinuously differentiable function " : G defined by ) 0 ¼ "i; jþ1 ; j 2 f0; . . . ; i 1g ð72Þ 0 ¼ i ð"ðZ Þ; Z Þ; i 2 f1; . . . ; mg from (65). Coupling the relation between (30), (31), (32) and (34) with (65) and (72) leads to the origin eðZ Þ ¼ 0 as the equilibrium. (H3): One is always able to act on the eigenvalues of ð@L=@eÞð0; Z Þ by choosing the respective design parameters of the PID i 1 -parts of the control law appropriately: comparing (66) with (50) leads to ð@L=@eÞð0; Z Þ with the same structure of the matrices given in (51). Thereby i; j;k ¼ i; j;k , i, j 2 f1; . . . ; mg, k 2 f0; . . . ; j g in view of (55). Thus it can always be assured, that there is a > 0 such that the eigenvalues of ð@L=@eÞð0; Z Þ have real parts no greater than . & Remark 2: It is important to remark that in interpreting Z as the ‘slowly-varying’ input of the tracking error system (65), one has to assure two conditions for its stability: desired trajectories which are designed ‘not too fast’ and the negativity of the eigenvalues of ð@L=@eÞð0; Z Þ. Since Z is composed of the desired flat output and its derivatives, it is evident that the farther left the eigenvalues of ð@L=@eÞð0; Z Þ can be modified by the respective i; j;k , the faster the desired trajectories can be designed. Therefore one remarks that the two different tasks of choosing the velocity of the desired trajectory and of modifying the point-wise in time ‘poles’ of the closed loop system have to be balanced carefully. Remark 3: Moreover, when considering i; j;k ¼ i; j;k ðZ ðtÞÞ it is clear that the poles of ð@L=@eÞð0; Z Þ can be placed to satisfy a desired characteristic polynomial.
547
Exact feedforward linearization
Φs
Is
Figure 3.
Equivalent circuit of a DC drive.
This could be interpreted as a dynamical kind of pseudo-linearization (Reboulet and Champetier 1984) around the desired trajectory or as a desired flatness based gain scheduling. Remark 4: Furthermore, in the context of Proposition 2, stability can also be analysed for partial state feedback by setting the respective control coefficients i; j;k ¼ 0, j ¼ ki;l þ 2; . . . ; j (as in (57)). The minimal number of derivative actions ki;l to be necessary (but not implicitly sufficient) for stability can be determined from (51), (54) and (55) using the necessary condition for negativity of the eigenvalues, that is the negativity of the coefficients of the characteristic polynomial of ð@L=@eÞð0; Z Þ. 8.
A separately excited DC drive example In this section, the theory developed in }} 3–7 is applied to a separately excited DC drive example. Figure 3 depicts the equivalent circuit of this machine (cf. Leonhard (1996)). A reduced order model taking flux saturation into account reads{ as J
d! c ¼ F ðU cFs !Þ B! l dt Rr s r
dFs ¼ Us Rs gðFs Þ dt
ð73Þ ð74Þ
where ! denotes the angular velocity of the shaft. The rotor and stator voltages are expressed by Ur and Us , the corresponding resistances by Rr , Rs . The stator flux is represented by Fs and is related to the stator current Is by the non-linear relation of the saturation effect Fs ¼ f ðIs Þ. The function f ðIs Þ is shown in figure 4, its inverse is denoted in (74) as Is ¼ gðFs Þ ¼ f 1 ðFs Þ. The parameter J expresses the
{ The dynamics of the rotor current Ir is neglected by a singular perturbation argument for the sake of simplicity.
Figure 4.
Typical magnetization curve of a DC motor.
moment of inertia of the rotor, B the viscous friction coefficient, c represents a constant dependent of the spatial architecture of the drive. The load torque l represents an unknown exogenous perturbation, it is dealt with in this section in order to give a first glance on how exogenous perturbations enter the respective equations. In Hagenmeyer et al. (2000) the flatness of the DC drive was studied intensively. It was shown that ½z1 ; z2 T ¼ ½!; Fs T denotes a flat output. Using the notation of (26), it is obvious that this equation ½1;1 ; 2;1 T ¼ ½!; Fs T also represents the state transformation of (27). Then the so-called Brunovsky´ state of (28) reads 9 1 c = _1;1 ¼ 2;1 ðUr c1;1 2;1 Þ B1;1 l > J Rr ð75Þ > ; _2;1 ¼ Us Rs gð2;1 Þ Thus, equation (24) is given by !9 2 > Rr c 2 = Ur ¼ J _1;1 þ B1;1 þ l þ 1;1 ð2;1 Þ > c2;1 Rr ð76Þ > > ; Us ¼ _2;1 þ Rs gð2;1 Þ The singularity of this equation for 2;1 ¼ Fs ¼ 0 represents the simple physical fact, that the DC drive can only operate when the stator flux (and therefore the stator current) is non-zero. In Hagenmeyer et al. (2000) desired trajectories were developed which take into account minimization of energy dissipation. Furthermore the aforementioned singularity is obviated in the cases of the starting of the drive and its braking to rest. These desired trajectories are used in the following as the a priori for the
548
V. Hagenmeyer and E. Delaleau
development of a controller based on exact feedforward linearization. Then the corresponding nominal feedforward u of (25) can be calculated as (considering the unknown nominal value{ of the load torque to be l ¼ 0) !9 2 > R c r 2 > Ur ¼ J _1;1 þ B1;1 þ 1;1 ð2;1 Þ = c2;1 Rr ð77Þ > > ; Us ¼ _2;1 þ Rs gð2;1 Þ The initial condition of a DC drive at rest is normally well known, that is ½1;1 ð0Þ; 2;1 ð0ÞT ¼ ½!ð0Þ; Fs ð0ÞT ¼ ½0 rad s1 ; 0 WbT . Thus, Proposition 1 states in its first part, that if (77) is applied to the system of (73) and (74) (or equivalently to (75)), then this system is equal for all times to the following linear system in multi-variable Brunovsky´ form (see (39)) _ 1;1 ¼ _1;1
ð78Þ
_ 2;1 ¼ _2;1
ð79Þ
if the given initial condition ½1;1 ð0Þ; 2;1 ð0ÞT ¼ ½0 rad s1 ; 0 WbT is considered for the design of the desired trajectory and also l ¼ 0. The second part of Proposition 1 assures that unique solutions of the non-linear system (75) exist around the desired trajectory n ðtÞ. Therefore, it is proposed in } 4 to take the Brunovsky´ form into account for the design of a combination of the nominal feedforward part with a feedback part, which in the DC drive example is used to track the desired trajectory n ðtÞ of the system in view of load torque perturbations l . When considering for the feedback part a PID-like controller as described in } 6, the whole control structure is given by (58) and reads for the DC drive example generally as 9 1 X > Rr > > Ur ¼ > þ 1;1; j e1; j J _1;1 > > c2;1 > j¼0 > > > > ! !> > 1 2 X > c 2 > > = þ 1;2; j e2; j þ B1;1 þ 1;1 ð2;1 Þ > R r j¼0 ð80Þ > > ! > > 1 1 > X X > > > þ 2;2;z;j e2; j þ 2;1; j e1; j Us ¼ _2;1 > > > > j¼0 j¼0 > > > > > ; þ Rs gð2;1 Þ
where the error e ¼ n n is defined as in (43) and the integrals of the feedback part { In Hagenmeyer et al. (2000), this value is observed online by a rapidly converging load torque observer which is omitted here for the sake of simplicity.
ei;0 ¼
ðt
ei;1 ðÞ d;
i 2 f1; 2g
0
as in (56). Considering the application of this control (80) to the system (75), the tracking error dynamics can be established as in } 5. For the given example it reads as 9 e_1;0 ¼ e1;1 > > > > > 1 X > e2;1 þ 2;1 > > _ > 1;1; j e1; j e_1;1 ¼ J 1;1 þ > > J2;1 > j¼0 > > ! !> > > 1 X > c2 2 > > þ 1;2; j e2; j þ B1;1 þ 1;1 ð2;1 Þ > > > R > r > j¼0 > > > > 2 = c 2 ðe1;1 þ 1;1 Þðe2;1 þ 2;1 Þ ð81Þ JRr > > > > > B > > ðe1;1 þ 1;1 Þ l _1;1 > > J J > > > > > e_2;0 ¼ e2;1 > > > ! > > 1 1 > X X > > > 2;2; j e2; j þ 2;1; j e1; j e_2;1 ¼ _2;1 þ > > > > j¼0 j¼0 > > ; þ Rs gð2;1 Þ Rs gðe2;1 þ 2;1 Þ The linearized tracking error (cf. the equation (51)) is then 2 0 1 6 6 1;1;0 1;1;1 e_ ¼ 6 6 0 0 4 2;1;0 2;1;1
system around the origin given by 3 0 0 7 1;2;0 1;2;1 7 7 e ð82Þ 0 1 7 5 2;2;0 2;2;1
where 9 > > > > > 2 2 > > c ð2;1 Þ B > > ¼ 1;1;1 > > > J JRr > > > > > ¼ 1;2;0 > > > 2 > _1;1 > B1;1 c 1;1 2;1 > > = ¼ 1;2;1 þ þ 2;1 J2;1 JRr > > > > ¼ 2;1;0 > > > > > ¼ 2;1;1 > > > > > > ¼ 2;2;0 > > > > > dg > > > ¼ 2;2;1 Rs > ; d2;1
1;1;0 ¼ 1;1;0 1;1;1 1;2;0 1;2;1 2;1;0 2;1;1 2;2;0 2;2;1
ð83Þ
2;1
After realizing that by respecting the general form of the control structure (58) unnecessary coupling of the subsystems was introduced for the case of the DC drive example, the following coefficients are set
Exact feedforward linearization
Figure 5.
549
Desired trajectories created by the nominal input (consistent initial condition).
1;2;0 ¼ 2;1;0 ¼ 0 s2 and 1;2;1 ¼ 2;1;1 ¼ 0 s1 . Thus (82) becomes an upper triangular block matrix 3 2 0 1 0 0 7 6 0 1;2;1 7 6 1;1;0 1;1;1 7 e 6 e_ ¼ 6 ð84Þ 0 0 1 7 5 4 0 0 0 2;2;0 2;2;1 Its characteristic polynomial reads (taking (83) into account) ! ! 2 c2 ð2;1 Þ B 2 s 1;1;0 s 1;1;1 J JRr ! ! dg 2 s 2;2;1 Rs s 2;2;0 ð85Þ d2;1 2;1
that is, the subsystems are decoupled with respect to stability. Given the necessary and sufficient condition for negative eigenvalues of s2 þ a1 s þ a2 is a1 ; a2 > 0, it can be deduced for (85) that even 1;1;1 ¼ 2;2;1 ¼ 0 s1 leads to stability, which shows the passivity property of the DC drive (remember that 1;1;0 and 2;2;0 stem from the introduction of the integrals in the controller, setting them to zero places obviously two poles at the origin— but also makes the use of the integrals ineffective). To enhance the convergence behaviour 1;1;0 ; 1;1;1 ; 2;2;0 ; 2;2;1 < 0
can be arbitrarily chosen with respect to their absolute Þ, i 2 f1; 2g, value. When considering i;i;1 ¼ i;i;1 ð2;1 then obviously the poles of the characteristic polynomial around the desired trajectory (85) can be placed (which could be interpreted as a dynamical kind of pseudolinearization (Reboulet and Champetier 1984) around the desired trajectory or a desired flatness based gain scheduling). Therefore the hypotheses of Proposition 2 are met and thus Proposition 2 guarantees stability (or asymptotic stability depending on the behaviour of the desired trajectory n ðtÞ for t ! 1). The parameters used for the simulations correspond to those identified on a real DC motor: Lr ¼ 9:61 mH, Ls ¼ 45 H, Rr ¼ 0:69 , Rs ¼ 120 , J ¼ 11:4 103 kg m2 , c ¼ 0:021 s.i., B ¼ 0:0084 Nm/rad. The saturation curve has been approximated by with Ls Io ¼ 19 Wb and f ðIs Þ ¼ Ls Io tanh ðIs =Io Þ Io ¼ 2:63 A. In figure 5, the results of the application of the desired input (77) corresponding to the desired profiles of the angular velocity and the stator flux to the DC drive (see (73) and (74)) are depicted in the case of a consistent initial condition. Note that the system perfectly follows the desired profiles without any error. A non-consistent initial condition for the stator flux of Fs ð0Þ ¼ 2 Wb (which could be interpreted as an initial stator flux resulting from a hysteresis effect) is taken into consideration for the results shown in figure 6 when applying the same nominal input (77) to the system
550
V. Hagenmeyer and E. Delaleau
Figure 6.
Application of the nominal input starting from a non-consistent initial condition.
(73) and (74) as in figure 5. It is clarified that there exists a unique solution in the vicinity of the desired trajectory, moreover the passivity property of the DC drive can be seen by the convergence of the resulting trajectory to the desired profile. Finally, the load torque presented in figure 7 is applied to the DC drive; its maximal value of 3:5 Nm is about half of the nominal load torque. To stabilize the desired trajectory, the control structure of (58) including the PI feedback part avoiding unnecessary coupling (1;2;0 ¼ 2;1;0 ¼ 0 s2 and 1;2;1 ¼ 2;1;1 ¼ 0 s1 ) with 1;1;0 ¼ 900 s2 , 1;1;1 ¼ 60 s1 , 2;2;0 ¼ 100 s2 and 2;2;1 ¼ 20 s1 is applied to the system, which gives the results depicted in figure 8. Thereby the initial condition of the stator flux is set to Fs ð0Þ ¼ 6 Wb, which can be seen as an involuntary premagnetization of the machine. Figure 8 shows that the overall control structure for the relatively slow maximal desired angular velocity illustrates the performance of exact feedforward linearization based on differential flatness combined with a simple PI control for each subsystem. 9.
Improving an existing PID control by exact feedforward linearization
The aim of this section is to illustrate the improvement of an existing PID control by adding to it a feedforward linearizing nominal control based on differential flatness. The chosen system consists of a small pendulum
which can levitate in the magnetic field created by a coil. The current of the coil can be regulated and plays the role of the control input. The setup can be controlled either by an analog PID controller or by a computer equipped with an acquisition card. The PID allows only stabilization of very small domains of approximately 0.1 cm. A reduced model{ for the magnetic levitation system is given by (see for example Le´vine, Lottin and Ponsart (1996)) 9 x_ 1 ¼ x2 > = 2 ð86Þ k i ; g > x_ 2 ¼ m c x1 Thereby x1 and x2 denote the position and the velocity of the pendulum respectively, i the electrical current as the input, m the mass of the pendulum, g the gravity acceleration constant. The parameter c stands for the nominal air gap and the parameter k depends on the permeability of the air gap, the reluctance of the core etc. As a matter of fact, the motion of the pendulum is naturally forced to maintain x1 < 0 as the magnetic coil is positioned at x1 ¼ 0. One flat output of the system is { The dynamics between the voltage and the current of the electric circuit can be neglected by a fast ‘nested’ control structure and a singular perturbation argument.
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Exact feedforward linearization 4
3.5
3
2
l
τ [Nm]
2.5
1.5
1
0.5
0
0
1
2
3 time [s]
Figure 7.
Figure 8.
4
Applied load torque.
Result of the closed loop system.
5
6
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Figure 9.
Response of the analog PID to position steps.
z ¼ x1 . Via the diffeomorphism (which in this case is simply the identity) ½1 ; 2 T ¼ ½x1 ; x2 T
ð87Þ
the preceding normal form can be rewritten in flat coordinates 9 _1 ¼ 2 > = 2 ð88Þ k i ; g > _2 ¼ m c 1 Therefore the control can be deduced in this case as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u 2 X u m _ i ¼ ðc 1 Þt j ej þ g ð89Þ 2 þ k j¼0 with the error e1 ¼ 1
and
e0 ¼
ðt 0
e1 ðÞ d
The following values are taken from a real magnetic levitation system.{ The nominal air gap is c ¼ 0:11 cm and the gravity acceleration constant is 981 cm/s2 . The unknown parameters can be identified as m ¼ 0:0844 kg and k ¼ 58:042 cm3 kg/(A s)2 . Three different experiments are presented for the purpose of comparison: 1: The first one consists in applying 0.2 cm position steps to the system controlled by the analog PID controller. The coefficients of this analog controller were tuned by the furnisher of the setup in an optimal way. The results of this case are reported in figure 9. One observes that the pendulum in levitation is stabilized { The experimental set-up is an Amira ma401 model (http://www.amira.de) located at the Universite´ Paris-sud. All the experiments have been conducted under the RTiCLab environment (http://rtic-lab.sourceforge.net) running RTLinux (http://www.rtlinux.org).
Exact feedforward linearization
Figure 10.
553
Response of the analog PID to a smooth position reference trajectory.
by the analog controller, however there is a poor transient response: The pendulum touches the constraints at x1 ¼ 0:01 cm and x1 ¼ 0:48 cm (mechanical stops) and the 5% response time is around 0.65 s. It is therefore not evident if the PID is able to stabilize the system for 0.2 cm steps without the mechanical stops. 2: The results of the second experiment are presented in the following figures. In figure 10 the response of the system with the same tuning of the analog PID controller is depicted in the case in which a smooth reference position trajectory (dotted line) is applied. One observes a far better transient response (the pendulum does not touch the stops anymore): the maximum error is reduced and the response time remains the same. The duration of the smooth transition is 250 ms. In figure 11 the response to the same type of smooth trajectory with a transition time of 500 ms is presented. In this case the maximum error is reduced (around 0.9 mm) and the response time is around 0.66 s. This second experiment illustrates the
advantages obtained by applying smooth trajectories to an existing PID controller. Note that the depicted smooth trajectories are not obtained by filtering. They are directly calculated from splines with the required regularity. 3: For the third experiment, the feedforward linearizing control law (89) is applied as designed above. The response is given in figure 12. The gains of the controller are 1 ¼ 30 000 ð1=s2 Þ, 0 ¼ 90 000 ð1=s3 Þ and 2 ¼ 400 ð1=sÞ. Since the PID part of the control law (89) enters non-linearly in the equation and its coefficients are parameterized by the desired trajectory, a comparison of the controller coefficients i with the analog PID gains is not possible. Nevertheless, in the case of the third experiment the error remains almost everywhere smaller than 5 % of the step between the low and high rest positions. The truly non-linear ‘feedforward linearizing controller’ is even better than the analog controller with smooth trajectories.
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Figure 11. Response of the analog PID to a smooth slower reference position trajectory.
10.
Conclusions
The trajectory aspect of differential flatness can be used to linearize a flat system by feedforward if the initial condition is consistent with the one taken into consideration for the nominal feedforward design. This might also clarify the difference between differential flatness and exact feedback linearizability. The new notion of exact feedforward linearization based on differential flatness furthermore leads to a specific control design to stabilize deviations from the desired trajectory: the resulting control structure consists of a combination of the nominal feedforward part and a feedback part. If extended PID control for the feedback part is considered, stability can be proven using a result by Kelemen. This control strategy is illustrated via a DC drive example. The experimental results of a magnetic levitation system show the importance with respect to practical applications of the presented control strategy: in the
case of a given flat non-linear system, for which there already exists a linear PID-like controller stabilizing the system in the vicinity of an operation point, a non-linear nominal feedforward based on flatness combined with a PID-like controller can achieve very good tracking or set point changes. In forthcoming publications robustness with respect to parametric uncertainty and exogenous perturbations will be established and aspects of specific applications of the presented control strategy will be investigated (see, for instance, the cascaded control structure presented in Hagenmeyer et al. 2002). Acknowledgements The authors are very grateful to Richard Marquez for many important discussions concerning the philosophy of this article. They are moreover thankful to Thomas Meurer, Michael Zeitz and Paulo Se´rgio Pereira da Silva for their respective advice regarding
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Exact feedforward linearization
Figure 12.
Response of the exact feedforward linearizing nominal control combined with PID stabilization.
the proof of Proposition 1. Veit Hagenmeyer was financially supported by the German Academic Exchange Service (DAAD). Emmanuel Delaleau is on temporary leave from Universite´ Paris-Sud, France. Appendix.
The Brunovsky´ state
The construction of a Brunovsky´ state was presented in Delaleau and Rudolph (1998, Proof of Theorem 1) in the intrinsic language of filtrations. It is chosen here to give another presentation mainly based on the mathematical framework of differentials in order to give an access to a reader not familiar with differential algebra. In the sequel, the differential of the variable a is denoted by da. Moreover, db ¼ ½db1 ; . . . ; dbr T represents the set of differentials of a finite set of variables b ¼ ½b1 ; . . . ; br T . Recall that u ¼ ½u1 ; . . . ; um T is the input of the considered flat system. The infinite dimensional vector space generated by the differentials of all the derivatives of the
control is denoted by U ¼ span fduðkÞ j k 2 Ng. In the same manner, the notation Z ¼ span fdzðkÞ j k 2 Ng is used in the following, where z ¼ ½z1 ; . . . ; zm T is a flat output. The algorithm is as follows: . Choose zo z such that the canonical image of dzo in ðspan fdz þ UÞ=U is a basis. . Then, for each r 0, choose zr z_ r1 such that the canonical image of dzr in ðspan fdz; d_z; . . . ; dzðrÞ g þ UÞ ðspan fdz; d_z; . . . ; dzðr1Þ g þ UÞ . is a basis. The choice zr z_ r1 is possible because it is evident that span fdz; d_z; . . . ; dzðrÞ g þ U ¼ span fdz; d_z; . . . ; dzðr1Þ ; dz_ r1 g þ U
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As the dimension of Z=U is finite (Flies et al. 1994), the algorithm terminates in a finite number of steps and the union of all the sets zr , r 0, is a state of the considered system. This state can be written: ð 1 1Þ
n ¼ ðz1 ; z_ 1 ; . . . ; z1
m 1Þ ; z2 ; . . . ; zð Þ m
ð1Þ
with the convention that zi ¼ 1. The i s are defined ðrÞ as i ¼ min fr 2 N j dzi 62 zr g. Finally n is a Brunovsky´ state of the considered flat system. References Brunovsky¤ , P., 1970, A classification of linear controllable systems. Kybernetika, 3, 173–188. Cazaurang, F., 1997, Commande robuste des syste`mes plats. Application a` la commande d’une machine synchrone. The`se de doctorat, Universite´ de Bordeaux I (France). Chelouah, A., Delaleau, E., Martin, P., and Rouchon, P., 1996, Differential flatness and control of induction motors. In P. Borne, M. Staroswiecki, J. P. Cassar and S. El Khattabi (eds) Proc. Symposium on Control, Optimization and Supervision; Computational Engineering in Systems Applications IMACS Multiconference, Lille, France, pp. 80–85. Delaleau, E., and Rudolph, J., 1998, Control of flat systems by quasi-static feedback of generalized states. International Journal of Control, 71, 745–765. Fliess, M., 1990, Generalized controller canonical forms for linear and nonlinear dynamics. IEEE Transactions on Automatic Control, 35, 994–1001. Fliess, M., Le¤ vine, J., Martin, P., and Rouchon, P., 1994, Nonlinear control and Lie–Ba¨cklund transformations: Toward a new differential geometric standpoint. In Proceedings of the 33th IEEE Conference on Decision Control, Lake Buena Vista, FL, USA, pp. 339–344. Fliess, M., Le¤ vine, J., Martin, P., and Rouchon, P., 1995, Flatness and defect of nonlinear systems: introductory theory and examples. International Journal of Control, 61, 1327–1361. Fliess, M., Le¤ vine, J., Martin, P., and Rouchon, P., 1999, A Lie–Ba¨cklund approach to equivalence and flatness of nonlinear systems. IEEE Transactions on Automatic Control, 44, 922–937. Fliess, M., and Marquez, R., 2000, Continuous-time linear predictive control and flatness: a module-theoretic setting with examples. International Journal of Control, 73, 606–623. Hagenmeyer, V., and Delaleau, E., 2002, Exact feedforward linearization based on differential flatness: the SISO case. In A. Zinober and D. Owens (Eds) Nonlinear and Adaptive Control, Vol. 281 of Lecture Notes Control Information Science (London: Springer), pp. 161–170. Hagenmeyer, V., Kohlrausch, P., and Delaleau, E., 2000, Flatness based control of the separately excited DC drive. In A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek (Eds) Nonlinear Control in the Year 2000 (Vol. 1), Vol. 258 of Lecture Notes Control Information Science (London: Springer), pp. 439–451.
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