Bond Graph Analysis for Control Design

Copyright @ IFAC Mechatronic Systems, Darmstadt, Germany, 2000

BOND GRAPH ANALYSIS FOR CONTROL DESIGN C. Sueur, G. Dauphin-Tanguy

Laboratoire d'Automatique et d'Informatique IndustrieUe de LiUe (LAIL UPRESA 8021) Eco/e Centra/e de LiUe, BP 48, 59651 Villeneuve d'Ascq cedex, France fax : +33-3-20-33-54-18, emai/ : sueur:ii;ec-lille. (r, gel! ii;ec-lille fr

Abstract : The aim of the paper is to show how a bond graph model may be used for analysis of structural properties, i.e properties depending only on the model structure and on the type of elements composing it, but not on the numerical values of the parameters. The properties pointed out in this way are generic, and can be used for designing some control laws with specific aims (as input-output decoupling, pole placement, disturbance rejection). The used methodology lies on causal manipulations on the bond graph model. The proposed procedure is implemented on a car suspension model. Different control strategies are proposed, discussed and compared (force or velocity actuators, accelerometers or tachymetcrs sensors, state or measurement feedback, complete or reduced model for control designing). Copyright @2000 IFAC Keywords: bond graph, structural properties, analysis, control design, symbolic calculus, input-output decoupling, disturbance rejection, car suspension, integrated design

1.

INTRODUCTION

properties which depend only on the model architecture and the type of physical phenomena retained as important in the modelling phase, and not on the numerical values of the parameters involved in their characteristic laws (except perhaps for few of them). After a first section presenting the different stages of the integrated design procedure we propose, a second section shows up its implementation on a guideline example which is an half vehicle mechanical suspension in heave and roll motion. The objective is the symbolic determination of control laws for inputoutput decoupling, pole placement and disturbance rejection. Different control strategies are analysed, diseussed and compared (force or velocity actuators, accelerometers or speedometers sensors, state or measurement feedback, complete or reduced model for control designing). Some of them are not implementable. For the others, symbolic control laws are calculated, and the corresponding close loop models determined formally and compared. Simulation results allow energetic and behavioural considerations and comparisons.

Two main points are specific in a mechatronic approach for the designing of controlled systems: - The procedure has several steps involving people specialised in different physical domains (mechanics, hydraulics, electronics, ... ). - Integrated design (a part of concurrent engineering) is based on a multidisciplinary team organised on a project basis. The same group is in charge of the complete design process, from the definition of the limits of the system under study (in terms of its environment) and from its objectives to the implementation of control laws on prototypes. This process includes modelling, analysis, architectural choice of the sensors and actuators and safety consideration. To achieve success, a crucial factor is the choice of a model usable during all the steps of this design procedure. The model needs to be at the same time a knowledge model allowing real physical insight and a representation model for controller design and validation. The goal of this paper is to demonstrate that the bond graph tool is well adapted for that purpose. One of the most difficult steps in the model design phase is the parameter identification, which is the main responsible of imprecision crrors. To deal with structural properties means to point out some

2. NOTATIONS AND RECALLS The goal is to recall the main tools for input-output dccoupling and disturbance rcjection with state or measurement feedback on bond graph models.

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Consider the perturbed system defined by equation (1) where x(.) EX"", Rn denotes the state vector, Y(.) E Y "'" R m denotes the vector of outputs to be

controlled.,

z( .) E Y "'" RP

=R

The state equation of the closed loop modeL in case of a measurement feedback, can be written as :

input vector, d(.) Er"", RP denotes the vector of disturbance inputs and A: X ~ X. B: U ~ X . C : X ~ Y are linear maps.

X.,_=A.,X., +J.,v + D.,d { Y -C.,xo,

y=Cx

A = ., [

(I)

z=Ex

G(s) = ~~:; = C(sI- A)- ]B,

= y(s) = C(sI-A) - ]D. des)

• Problem formulation for input output decoupling Find a state feedback (2) or a static measurement feedback (3) or a dynamic measurcment feedback (4) such that the closed loop transfer matrix between the outputs to be controlled and the control inputs is diagonal (when the system is supposed to be non perturbed).

m {u = Lw+Kz +Jv

A+BKE ME

It is important to notice that matrices F and G are obtained when solving the problem with state feedback (matrix F) and with measurement injection (matrix G). It means that these matrices contain parameters for pole placement. A useful property is that the set of poles in the closed loop model is the union of the set of poles of matrices A + BF and A + GC, that is a-(Ao» =a-(A + BF) u a-(A + GC) .

The transfer functions used in the following are :

u = Fx+Jv u=Kz+Jv =Nw+Mz

(9)

with

X =Ax+Bu +Dd

H(s)

BS +D

denotes the vector of

measured outputs, u (.) E U "'" R m denotes the control

{

= In'

and matrix L2 is calculated with subspaces defined in the geometric approach (Arib et al., 2000). v is the new control variable. L]E +L2

When a static state feedback can be used (for input output decoupling or disturbance rejection), an estimator must be built if the state is not available for measurement. The measured output can be used in this case. The control structure is presented figure 1. Three approaches can be used to solve these problems : the geometric approach, the structural approach (based on the input output description), the bond graph approach.

(2) (3) (4)

The concept of infinite structure is used., and some specific parameters are defined.

•

Problem formulation for disturbance rejection (with perturbation measurement ifnecessary) Find a state feedback (5) or a static measurement feedback (6) or a dynamic measurement feedback (7) such that the closed loop transfer matrix between the outputs to be controlled and the disturbance inputs is null. u = Fx+Jv+Sd u=Kz+Jv+Sd = Nw +Mz +Rd

m {u = Lw+Kz+Sd +Jv

z

(5) (6)

Fig. 1 . Control structure with state estimator (state feedback)

(7)

with N ~ A + GE + BFL2 M -BFL]-G {

(8)

L=FL 2 K=FL]

In equations (5-7), the matrices Rand S are equal to zero if perturbation measurement is not necessary. The matrix S is the same for each control strategy.

Fig. 2 . Control structure with compensator (measurement feedback)

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They are: g, : infinite structure of subsystem (ciA,B) : length of the shortest causal path between the i th output to be controlled and the set of control inputs g; : global infinite structure of system (CA,B) hij : infinite structure of subsystem

- the road motion (on left and right wheels) and the gravity which are taken into account by the passive suspension and the wheels - the driver actions (the mass transfer force due to braking or accelerating effects applied on the car inertia center G and the torque due to steering) which have to be compensated by the activation.

(ciA,D})

length of the shortest causal path between the i th output to be controlled and the jth disturbance input

The corresponding bond graph model, including the modelling hypotheses, is given figure 4.

n~ :column infinite structure of subsystem (EA,DJ )

: length of the shortest causal path between the jth input disturbance and the set of measured outputs

3.1 Analysis for designing control laws

The different analysis steps for the designing of control laws are:

The following properties are given without guarant)ing the stabilit)· propert)· of the closed loop system. With the bond graph approach, the analysis is applied on the bond graph model with a derivative causality assignment using the same concept of input output causal paths. Index d is added when the causal paths are defined on the bond graph model with a derivative causality assignment (ex. g,d)' Structural Conditions for input-output decoupling wi th state feedback • The model is invertible (the number of disjoint input-output causal paths is equal to m) • The row infinite structure is equal to the global infinite structure (the shortest causal paths are disjoints or {g i } == {g; } )

1-

StudY of the passive model structural properties: dimension of the state vector, model order. structural ranks of state equation matrices

2-

Definition of the objectives for the system and constraints for the controller : modi1)r the dynamics by pole placement reject the disturbances due to the driver actions

3- Determination of the possible strategies. and the corresponding sensor and actuator architecture : different types of feedback : state feedback or measurement feedback? control design on the global model or on a reduced model? different types of actuators : force actuators or velocity actuators? different types of sensors : force sensors or speed sensors?

Structural Conditions for disturbance rejection when the disturbance is available for measurement (without stability) - Structural conditions: rank{G(s)] == rank{G(s) , H(s)] { L oo [G(s)] == L oo [G(s),H(s)] (10)

- Bond graph conditions • the number of disjoint input-output paths is the same for the bond graph model with and without the disturbance (the disturbance is considered as a control input) • the infinite structure (defined with the shortest disjoints input-output causal paths) is the same with and without the disturbance inputs.

z

Necessary Structural Condition for disturbance with measurement feedback (without closed loop guarantied stability) (11)

3. IMPLEMENTATION ON AN EXAMPLE The studied model is an 112 vehicle passive mechanical suspension (fig. 3.), supposed to behave in the lincar part of the suspension and tire springs characteristic laws. The considered motions are heave and rolL The disturbance inputs are:

Fig. 3 Bicycle suspension - the general scheme and the physical representation with components 4- Implementation of the different strategies and structural analysis of the feasibility of the controller:

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First conclusion: this model is decouplable with static state feedback Using a control law defined with these causal paths, non unassigned modes. there are seven (9 - L These modes are the invariant zeros of the model (C,A, B) . The null invariant zeros are pointed out on the bond graph model in derivative causality (fig.5.) using input output causal paths: Shortest input output causal paths gid :

some structural properties have to be verified on the complete active model with its control and measurement architecture. They are: measurability of the considered variables (Sueur, et al., 1991» invertibility of the model (Bertrand, et of. , 1997a). input-output decouplability (Bertrand, et 01. , 1997a). characterisation of the infinite and finite structures (Bertrand, et al., 1997b, Van der Woude, 1991) capability for the disturbance to be rejected, with or without its measurement (Fabien, et al., 1975, Arib, et aI. , 2000) 1-0 decoupling and disturbance rejection with stability (existence of unstable fixed modes) (Bertrand, et 01. , 1997b, Martinez Garcia, et 01 1994)

g;)

(c) ,A, B) Df*(VG)-Csz -MSe(Factz ) : (cz,A, B) Df*(wG) -TF -Cs) -MSe(Factl ) : gld

= 1,

gZd

= 1 and

gid

= g;d = 1 .

Second conclusion : There are two null invariant zeros. As {gid} = {g;d } , and because the other invariant zeros are stable, a stable closed loop system with five fixed modes can be defined.

Steps 1, 2, and 4 involve only causal manipulations on the bond graph model and study of causal paths between elements.

It can be remarked that the same properties are obtained on a reduced model built by suppressing in the bond graph model the fast dynamical elements associated with the tires and the unsprung masses, because whcn studying thc structural propertics, the same causal paths are defined. The advantages in using a reduced order model (order 5 rather than 9) appear in the size of the state estimator or compensator we have to construct.

3.2 Study of the passive model

From the bond graph model in integral causality (BGI) figure 4, we can deduce that : all the states variables are statically independent the model order is n = 9 the state model is a set of ODEs From the bond graph model in derivative causality (BGD), we can deduce that : the BG-rank of the state matrix is q = 9, thus no static mode exists. only one actuator and one sensor are needed for the model to satisfY structural controllability and observability properties. The choice of their placement is only dependent on technological considerations.

When studying disturbance rejection with static state feedback, it can be shown that the perturbation measurement is required, because the length of the causal path between the outputs to be controlled and the control inputs are same as the length of the causal path between the outputs to be controlled and ~he disturbance inputs. The control law allowmg disturbance rejection gives a decoupled model. The results are gathered in table 1.

3. 3 Study of the active model

Structural conditions for disturbance rejection by measurement feedback (speed sensors):

The variables to be controlled are the car inertia centre G velocities in vertical motion Vme and in

On the bond graph model fig.4., it can be shown that

rotational motion

h) I

WJ

o

.

Then two control variables

are needed for implementing input-output decoupling and disturbance rejection with pole placement. The study is proposed for force and velocity actuators, and two velocity sensors represented fig. 4 by Df.

= hzz = 1,

h12

= hZ) = 2

necessary relation

hif

~

and n{ = n~ = 1. The

gi + nj

for disturbance

rejection is nol always satisfied. First conclusion: Disturbance rejection cannot be achieved with the proposed speed sensors.

Structural conditions for input output decoupling with static state feedback (force actuators): Shortest input output causal paths length gi :

Nevertheless, disturbance rejection can be achieved with the measurement of the perturbation but the sensors are not useful. They are used here for pole placement of the closed loop system.

(c) ,A, B) Df*(VG)-I mo -TF - MSe(Factl ) : g ) =1 (cz, A ,B) Df*(wG) - I J ,.".. - TF-MSe(FactZ ) :

It can be shown in that case that matrix L2 in equation (8) is equal to the identity matrix, that is Lz = In . In that case, L) = 0 and it comes :

g 2 =1 gi = g; = 1 (The shortest paths are different paths).

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to the system designer a very easy to implement set of graphical procedures for performing structural analysis and symbolic control law determination. The main point we try to show up is that it is possible to make a deep study of the model properties and its control structure without any consideration on numerical values of parameters. The main restriction concerns the present limitation of the proposed properties to the linear case; a linearization has to be performed on the initial model, which is not always possible. Some of the procedures have been eA1ended to some particular forms of non linearities, but it is not generally extendable. The main constraint deals with the minimisation of the energy supplied by the actuators, which is not studied here.

N=A+GE+BF M=-G {

(12)

L=FL2

K=O Matrix F is the state feedback matrix used for disturbance rejection and matrix G can be chosen in any way. Matrix S is the same as with a state feedback and matrix R satisfies relation R = BS + D . It can be remarked that this matrix is equal to zero when force actuators are used, because it is possible to find a matrix S which satisfies the relation BS + D = 0 . The results are gathered in table 2. Table 1. Analvsis with state feedback Speed Actuator yes

invertibility I/O decouplability yes with stability Invariant 7 zeros stable Closed loop two 1I1!t order model order decoupled models driver perturbation rejection Closed loop model order with disturbance rejection

yes Perturbation measurement two 1I1!t order decoupled models

REFERENCES

Force Actuator yes

Arib 1., C. Sueur, G. Dauphin-Tanguy (2000) Disturbance rejection by measurement feedback for bond graph models. Proceeding of Mathmod 2000, Vienna. Austria, Vol. 2, pp. 709-714. February 2000. Bertrand J.M., C. Sueur, G. Dauphin-Tanguy (1997a) Bond graph for modeling and control structural analysis tools for the design of inputoutput decoupling state feedbacks. Proceedings of the ICBGM '97, part of the SCS Western Multiconference, Phoenix Arizona, USA, 103108. Bertrand J.M., C. Sueur, G. Dauphin-Tanguy (1997b) On the finite and infinite structures of bond graph models, Proceedings of the IEEE/SMC Conference , Orlando Floride, USA, vol 3, 24722477. Fabien E. and W.M.Wonham (1975) Decoupling and Disturbance Rejection, IEEE Transactions on Automatic Control, Vol. 20, 399-401. Martinez Garcia J.C. and M. Malabre (1994) The Row by Row Decoupling Problem with Stability: A Structural Approach, IEEE Transactions on Automatic ControlvoI. 39 (12), 2457 -2460. Sueur C. and G. Dauphin-Tanguy (1991) Bond graph approach for structural analysis of MIMO linear systems, J. Franklin Inst. , Vo1.328 (1), 55-70 Van Der Woude J.W. (1991) On the Structure at Infinity of a Structured System, Linear Algebra and its Applications, Vol. 148, 145-169.

yes 7 2 null, 5 stable two 2nd order decoupled models with a null zero at each numerator yes Perturbation measurement two 2nd order decoupled models with a null zero at each numerator

Table 2. Analysis with measurement feedback Speed Actuator invertibility yes 7 Invariant zeros stable driver yes perturbations Perturbation rejection measurement 18 Closed loop model order 7 stable fixed with modes disturbance rejection

Force Actuator yes 7 2 null, 5 stables yes Perturbation measurement 18 5 stable fixed modes

4. CONCLUSION The combined use of integral and derivative causality assignment and causal path and loop properties gives 653

Im;

!

A;

D

1

Facti

Sf ,"2

Sf""

Fig.4. Force actuator and bond graph model in integral causality assignment, with sensors (real as Dfregrouped in the measurement vector z and virtual as Df'I' regrouped in the non measurable outputs to be controlled vecteur y) and actuators (as modulated sources MSe) in(;

XOl.d

XOl.d

61

c...

c""

0 ~

1 "-- 0

61

1

Rta Sf ...

Sf:"ll

Fig.5. Bond graph model in preferred derivative causality assignment - force actuators hoG

A;

1 ~ 8

1 Sf",

•

~ 1 ,11.'

1L

Sf""

Fig.6. Velocity actuator and bond graph model in preferred integral causality assignment with control inputs as MSf

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