Input and state observer for LTV bond graph models

Input and state observer for LTV bond graph models Dapeng Yang, Christophe Sueur Ecole Centrale de Lille, LAGIS CNRS UMR 8219, BP 48, 59651, Villeneuve d’Ascq cedex, France [email protected] Abstract: The object of this paper is the study of input and state observers for linear bond graph models. The classical mathematical approaches are first recalled and a graphical approach which is based on causality, structural properties of bond graph models and some concepts of module theory is proposed. A particular attention is concerned with the study of invariant zeros which play an important role for the stability property of the system. An example with time varying parameters is proposed. Keywords: Unknown input observers, Bond graph, LTV models, Module theory, Invariant zeros. 1. INTRODUCTION Linear Time Invariant (LTI) systems have been intensively studied since fifty years at different levels, from modeling to control. The design of controllers requires the integration of many kinds of information and a first step is often dedicated to the analysis of the model. LTI models have structural invariants which play a fundamental role and the structure of LTI models has been extensively studied in many papers and books Basile and Marro (1973), Morse (1973), Rosenbrock (1970), Kailath (1980), Gilbert (1969), Kalman et al. (1969). Note that among these structural invariants, the knowledge of zeros (decoupling zeros, transmissions zeros, invariant zeros) is often an important issue because zeros are directly related to some stability conditions of the controlled system. The infinite structure of LTI models is often related to solvability conditions and is studied in Dion and Commault (1982), Descusse and Dion (1982) and a structured system approach has been proposed in Reinschke (1988), van der Woude (1991). The geometric approach, based on the concept of invariant subspaces has been developed in different books Wonham and Morse (1970), Basile and Marro (1992), Wonham (1985). The advantage is that this approach allows at the same time the study of the solvability conditions and the control synthesis for various control problems. The unknown input and state observability problem (UIO) is a well known problem because for control design with a state space approach, the state vector cannot be entirely measured (an observer must be designed) and the system is often subject to unknown inputs (disturbance or failure...) which must be estimated. Different approaches give solvability conditions and constructive solutions. For LTI models, constructive solutions with reduced order observers are first proposed with the geometric approach Guidorzi and Marro (1971), Bhattacharyya (1978), Basile and Marro (1973). Constructive solutions based on generalized inverse matrices taking into account properties of invariant zeros are given in Kudva et al. (1980) and then in Miller and Mukunden (1982) and Hou and Muller (1982) with observability and detectability properties. Full order observers are then proposed in a similar way (based on generalized inverse matrices) in Darouach et al. (1994) and Darouach (2009), but with some restriction on the infinite structure of the model. The algebraic approach is proposed in Trentelman et al.

(2001) and in Daafouz et al. (2006) for continuous and discrete time systems, without restriction on the infinite structure of the model. A graphical approach is in Boukhobza et al. (2007). The objective of this paper is the investigation of observers for linear systems modeled by bond graph Karnopp et al. (1975) and Rosenberg and Karnopp (1983) when the system has two kinds of inputs: measured an unmeasured inputs. The first section deals with the problem statement. In the following sections, procedures are proposed and solvability conditions are given for the UIO problem in case of bond graph models. In order to extend graphical procedures developed for LTI models, the algebraic approach is applied conjointly with the graphical approach. Notice that, at the analysis step, proposed methods on bond graph models do not require the knowledge of the value of parameters, because intrinsic solvability conditions can be given and that a formal calculus could be proposed at the synthesis level. An illustrative example of a DC motor is proposed. In this example, different choices for the unknown input variable (disturbance) illustrate the generic solvability. At the end, the module theoretic approach is justified by an extension to the LTV case, with some simulations. 2. UIO PROBLEM STATEMENT The classical state representation, for a linear time invariant system (LTI), is given by the Kalman form (1), with x ∈ ℜn and y ∈ ℜ p . The input variables are divided into two sets u ∈ ℜm and d(t) ∈ ℜq which represent known and unknown input variables respectively. It is supposed that input variables u(t) and d(t) are bounded and infinitely continuously derivable. { x˙ (t) = Ax (t) + Bu (t) + Fd (t) (1) y (t) = Cx (t) Models described in equation (1) without unknown inputs are denoted as Σ(C; A; B) in the LTI case and Σ(C(t); A(t); B(t)) in the LTV case. Models are supposed to be invertible, with full matrix rank for matrices C, B and F. 2.1 Existence conditions Much attention has been paid to the algorithms to compute observers, but firstly existence conditions must defined. The

concepts of strong detectability, strong* detectability and strong observability have been proposed in Hautus (1983). Theses concepts are useful for solving the UIO problem.

 ( )  x˙ˆ = (PA − LC) xˆ + Q y(r) −U + Ly + Bu ) ( ) (  dˆ = CAr−1 F −1 y(r) −CAr xˆ −U

System (1) (with only unknown input d(t)) is strongly detectable if y(t) = 0 for t > 0 implies x(t) → 0 with (t → ∞) and system (1) is strong* detectable if y(t) = 0 for t → ∞ implies x(t) → 0 with (t → ∞).

dˆ is the estimation of d and the matrices Q and P verify: Q = r−1 ( )−1 F CAr−1 F ; P = In − QCAr−1 , and U = ∑ CAi Bu(r−1−i) . r

The strong detectability corresponds to the minimum-phase condition, directly related to the zeros of the system (1) (without input u(t)) defined as to be the values of s ∈ C (the complex plane) for which (2) is verified. ( rank

sI − A −F C 0

)

( < n + rank

−F 0

(3)

i=0

is the infinite zero order between the unknown input d(t) and the output y(t). The main idea of the method is to implement derivations on the output variable y(t) to let the unknown input variable d(t) appears explicitly. The r-order derivation of output r−1

variable is: y(r) = CAr x + CAr−1 Fd + ∑ CAi Bu(r−1−i) , where

)

i=0

(2)

Proposition 1. Hautus (1983) The system Σ(C; A; F) in (1) is strongly detectable if and only if all its zeros s satisfy Re(s) < 0 (minimum phase condition). Proposition 2. Hautus (1983) The system Σ(C; A; F) in (1) is strong* detectable if and only if it is strongly detectable and in addition rank[CF]=rank[F]. Proposition 3. Hautus (1983) The system Σ(C; A; F) in (1) is strongly observable if and only if it has no zeros. An algebraic characterization, which is equivalent to the previous one is given in Daafouz et al. (2006) and Barbot et al. (2007). The LTI system Σ(C; A; F) in (1) can have an unknown input observer if, and only if, it is left invertible and it has the minimal phase property. The LTI system Σ(C; A; F) in (1), supposed to be asymptotically observable with unknown input, is rapidly observable if, and only if, it zero dynamics is trivial. Finally, a linear system is said to be observable with unknown inputs if, and only if, any system variable, a state variable or an input variable for instance, can be expressed as function of the output variables and their derivatives up to some finite order. In other words, an input-output system is observable with unknown inputs if, and only if, its zero dynamics is trivial and if moreover the system is square, it is flat. 2.2 Synthesis of the observer In order to solve the UIO problem for systems in (1), a necessary condition called observer matching condition for the existence of observers is often required (see Kudva et al. (1980); Darouach et al. (1994)): rank CF = rank F = q. However, this condition is not always satisfied. Floquet and Barbot (2006) proposed unknown input sliding mode observers after implementing a procedure to get a canonical observable form of systems. This method can also be extended in the nonlinear case. Daafouz et al. (2006) gave an intrinsic explanation of the UIO problem by the algebraic approach for systems in (1). A general structure of UIOs for estimating state and unknown variables was proposed with the previous mentioned necessary and sufficient conditions: system Σ(C; A; F) is left invertible and minimum phase. The first condition is equivalent to rank C(sI − A)−1 F = q. Most of the previous works do not take into account the control inputs. An extension is proposed here with these input variables. The unknown input observer for a SISO model with control input is written in (3).

CAr−1 F ̸= 0 and CAs−1 F = 0 for s < r. Note that the control input must be derivable (r − 1 times) which is possible for example with a flat control approach. For MIMO models, the extension of the procedure was proposed by Floquet and Barbot (2006). The dynamic of the estimation error of state variables is e˙ = x˙ − x˙ˆ = (PA − LC) (x − x). ˆ One has limt→∞ e(t) = 0 for any x(0), x(0), ˆ d(t) and u(t). The estimation of d can be written ( )−1 as dˆ = CAr−1 F CAr (x − x) ˆ + d. As limt→∞ e(t) = 0, then ˆ = d(t). limt→∞ d(t) The design of the observer (3) can be obtained as following: (1) Verify the minimum phase property of systems Σ(C; A; F). (2) Compute the relative degree r and the inverse of CAr−1 F. (3) Compute Q, P and U and then L for pole placement. 3. UOI FOR LTI BOND GRAPH MODELS In a bond graph model, causality and causal paths are useful for the study of properties, such as controllability, observability and systems poles/zeros. Bond graph models with integral causality assignment (BGI) can be used to determine reachability conditions and the number of invariant zeros by studying the infinite structure. The rank of the controllability matrix is derived from bond graph models with derivative causality (BGD). Systems invariant zeros are poles of inverse systems. Inverse systems can be constructed by bond graph models with bicausality (BGB) which are thus useful for the determination of invariant zeros. Some of these concepts are recalled in this part for LTI bond graph models. 3.1 Controllability/Observability The controllability/observability properties have been first derived from a graphical approach using the causality concept in Sueur and Dauphin-Tanguy (1991). A LTI bond graph model is controllable iff the two following conditions are verified: first there is a causal path between each dynamical element and one of the input sources and secondly each dynamical element can have a derivative causality assignment in the bond graph model with a preferential derivative causality assignment (with a possible duality of input sources). The observability property can be studied in a similar way, but with output detectors. 3.2 Infinite structure and invariant zeros The infinite structure of multivariable linear models is characterized by different integer sets. {n′i } is the set of infinite zero

orders of the global model Σ(C; A; B) and {ni } is the set of row infinite zero orders of the row sub-systems Σ(ci ; A; B). The infinite structure is well defined in case of LTI models Dion and Commault (1982) with a transfer matrix representation or with a graphical representation (structured approach), Dion and Commault (1993), and can be easily extended to LTV models with the graphical approach. The row infinite }zero order ni verifies condition ni = min { k|ci A(k−1) B ̸= 0 . ni is equal to the number of derivations of the output variable yi (t) necessary for at least one of the input variables to appear explicitly. The global infinite zero orders Falb and Wolovich (1967) are equal to the minimal number of derivations of each output variable necessary so that the input variables appear explicitly and independently in the equations. From a bond graph approach, the order of the infinite zero for the row sub-system Σ(ci ; A; B) is equal to the length of the shortest causal path between the ith output detector yi and the set of input sources. The global infinite structure is defined with the concepts of different causal paths (not recalled for sake of space). The number of invariant zeros is determined by the infinite structure of the BGI model. The number of invariant zeros associated to a controllable, observable, invertible and square bond graph model is equal to n − ∑ n′ i . For bond graph models, invariant zeros equal to zero can be directly deduced from the infinite structure of the BGD model Bertrand et al. (1997). Invariant zeros for LTV models have been studied in Yang et al. (2011). In that case the concept of bicausality associated to an algebraic approach is useful.

The bond graph model is controllable and observable (a derivative causality can be assigned). The numerical values of system parameters are shown in Table 1. From structural calculation, condition rank CF = rank F = q is satisfied, hence the observer proposed in Darouach (2009) can be used, but not proposed here. Table 1. Numerical values of system parameters L 5 × 10−4 H

R 0:25 Ω

k 1

J 5 kgm2

b 0.2 Nm/Wb

d(t) sin(t)

u(t) Γ(t)

The causal path length between the output detector D f : y and the disturbance input Se : d is equal to 1, path D f : y → I : J → Se : d, thus there is an invariant zero in the system Σ(C; A; F) and r = 1. After calculations or analysis of the bond graph model with a bicausal assignment, the invariant zero is s = − RL = −500 which verifies the minimum phase condition. Matrix CF is equal to 1. Matrix L is used to place poles of the observer. One pole is fixed (invariant zero of system Σ(C; A; F)), another is placed at s = −20. Through the procedure proposed in Section 2.2, the matrices of the observer are [ ] [ ] 0 −500 −0:4 Q= PA − LC = 5 0 −20 ] [ (5) 1 U =0 L= 100 The estimation errors of the two state variables with an initial condition x(0) = [2 1]T are displayed in Fig. 2. 4

3.3 UIO for BG models: example

e1 3

The design of the observer defined in (3) and proposed in the previous section can thus be redesigned from a bond graph approach. It will be directly proposed on the following example. An example of a DC motor is used to show the procedure for designing the UIO observer. The BGI model of the system with a disturbance signal is given in Fig. 1, and the state-space equations are presented in (4), with x = (pL ; pJ )t = (x1 ; x2 )t the state vector, y the measured output variable, u the control input variable and d the disturbance input variable. Here the disturbance input is supposed to be d(t) = sin(t), and the input u(t) is the Heaviside unit step function, i.e. u(t) = Γ(t).

e2

2 1 0 −1 −2

0

1

2

3

4

5

time

Fig. 2. Trajectories ei = xˆi − xi ; i = 1; 2 of system 1 The estimation of d is dˆ = 5(y˙ − [ 400 −0:008 ] x). ˆ The comparison of the estimation of the unknown input de (t) and itself d(t) is shown in Fig. 3. Now the output y(t) is replaced into the 1-Junction related to the element I : L, i.e. the output matrix is C = [ L1 0]. The bond graph model is controllable and observable. The condition rank CF = rank F = q is no more satisfied. The causal path length between the output detector D f : y and the disturbance input Se : d is equal to 2, i.e. r = 2, path D f : y → I : L → GY → I : J → Se : d, thus there is not any invariant zero and the model is flat (with input d and output y). It means that poles of the observer can be freely assigned. An unknown input observer can be used.

Fig. 1. BGI model of the DC moteur  R k    x˙1 = − L x1 − J x2 + u   k b x˙ = x − x + d  2 L 1 J 2     y = 1 x2 J

(4)

As r = 2, the unknown input can be represented by a 2order differential polynomial of the output y(t). One has CF = 0;CAF = −400. The matrices of the observer are:

4. EXTENSION TO LTV BOND GRAPH MODELS

25 de 20

For linear bond graph models with time varying parameters, the previous approach must be redefined. Indeed, much of the concepts cannot be used, such as poles and zeros which must be defined in an extended field. The structural approach on bond graph models must be used with the algebraic approach. Some of the concepts developed in different works are briefly recalled and then applied on an example, for the UIO problem.

d

15 10 5 0 −5

4.1 Algebraic approach 0

5

10 time

15

20

Fig. 3. Trajectories de (t) and d(t) of system 1 [

] [ ] −900 −0:2 0 PA − LC = −0:0025 2:45 × 106 500 [ ] 0:2 L= U = 2000u˙ − 1 × 106 u −600

Q=

(6)

The poles of the observer are arbitrarily placed by the matrix L, here s1;2 = −200. Fig. 4 shows the estimation errors of two sate variables of the second system with an initial condition x(0) = [0:1 2]T . 10 e1 e2

( I δ − A(t) −B(t) )

0

0

5

10 time

15

20

Fig. 4. Trajectories ei = xˆi − xi ; i = 1; 2 of system 2 The of d is ]dˆ = −0:0025(y(2) + 1 × 106 u − 2000u˙ − [ estimation 8 ˆ Trajectories of the estimation de (t) 4:992 × 10 200016 x). and the unknown input e(t) are shown in Fig. 5. 5 de d 0

−5

−10

In Fliess’s theoretic approach the systems are the modules. The definitions in this section are the same with the one introduced in Fliess (1990). A linear system ∑ is a finitely generated left R-module. Definition 1. Fliess (1990) A (linear) dynamics D is a system in which a finite set u = {u1 ; u2 ; :::; um } of input variables is such that the quotient module D/[u] is torsion. It means that any element in D can be calculated from u by a linear differential equation. In the LTV case, equation (1) without unknown input and output is equivalent to (7) in a module framework representation.

5

−5

The study of commutative rings and fields, provides good tools for understanding algebraic equations. Differential algebra Ritt (1950) and Kolchin (1973) uses differential equations concepts and results from commutative algebra. These concepts are nowadays used in the control community, because it is possible to treat in a similar way problems for classical linear systems and linear time varying (LTV) or non linear (NL) systems with different representations.

0

5

10 time

15

Fig. 5. Trajectories de (t) and d(t) of system 2

20

( ) ( ) x x = R(δ ;t) =0 u u

(7)

The entries of the matrix R(δ ;t) belong to the non commutative ring R = K[δ ] and [ x u ] ∈ Ω (here x and u are considered as row vectors, Ω is the R-module and δ = d/dt). The structural properties of (7) are then translated to a module framework. The controllability property is thus directly deduced from the Jacobson form of matrix R(δ ;t), or from the torsion module associated to the state equation, if this torsion element exits. The dynamics D is said to be observable iff D = [y; u] (Fliess 1990). In other words, D is observable iff every element of D can be expressed as a linear combination of the components of y and u and of their derivatives of any order. Definition 2. Fliess (1990) [y; u], viewed as a dynamics with input u and output y, is called the observable dynamics of D. Consider a LTV system Σ(C(t); A(t); B(t)) represented by equation (1) which is a finitely generated module M over the ring R = K[δ ]. The module Miz = T (M/[y]R ) is torsion and is called the module of invariant zeros of Σ, Bourl`e[ s and ] Marix¯ nescu (2011). The module Miz is defined by P (δ ;t) , where u¯ [ ] δ I − A (t) −B (t) P (δ ;t) = is the system’s matrix and x, ¯ u¯ C (t) 0 are the images of x, u in module Miz . P(δ ) is singular with certain δ = αi . With these values of δ , for an input u(t) =

u0 eα t ;t ≥ 0, there exist initial state variables x0 such that the output is null:y ≡ 0;t ≥ 0.

10 e1 e2

4.2 Bond graph approach

5

For non linear models, variational models can be written with Kahler derivation. These new models are linear time varying (LTV) models. Non linear bond graph models are transformed in LTV bond graph models with some graphical procedures proposed in Achir and Sueur (2005). In case of the stability analysis, the finite structure must be studied, and in that case the extension to the LTV case is not so easy. Different classical problems such as the controllability/observability analysis Chalh et al. (2007), Lichiardopol and Sueur (2010) or inputoutput decoupling problem Lichiardopol and Sueur (2006) are proposed as a direct extension of the LTI case due to some properties of the bond graph representation. In the LTV case, the design of the observer defined in (3) must be redefined, except for the part dealing with the structural analysis with the bond graph approach. The new equations are not written in this paper, due to space. It is directly proposed on the following example, with a particular attention to the problem of poles and zeros. Now, the first system in the above example is extended to an LTV system with a time-varying element GY : k(t), let k(t) = cos(t) + 1:2. Obviously, the conclusions for structural analysis in the LTI case are still valid for the LTV system. However, one should pay attention to calculation with timevarying parameter in matrices over the noncommutative ring of differential operators. System Σ(C(t); A(t); F(t)) contains one invariant zero, which is δ = −500. So the minimum phase condition is satisfied. Procedures are similar to the LTI case. The matrices of the observer are [ ] [ ] 0 −500 −0:24 Q= PA − LC = 5 0 −1 [ ] − cos (t) L= U =0 5

0

−5

0

1

2

3

4

5

time

Fig. 6. Trajectories ei = xˆi − xi ; i = 1; 2 of the LTV system The comparison of the estimation of the unknown input de (t) and itself d(t) is shown in Fig. 7. 25 de 20

d

15 10 5 0 −5

0

5

10 time

15

20

Fig. 7. Trajectories de (t) and d(t) of the LTV system (8)

In the LTV case, the finite structure may be influenced by time-varying parameter in system matrices as shown by Yang et al. (2011). Invariant zeros and poles of systems can be derived from Smith form of matrices over differential rings. A ˙ Let new rule called Leibniz rule is required, i.e. δ a = aδ + a. L(t) = [l1 (t) l2 (t)]t be the time varying matrix chosen for pole placement of the observer. The definition matrix of poles of the observer is [I δ − PA + LC]. The invariant polynomial of this matrix in case of the example is (δ + 0:2l2 (t))(cos(t) + 1:2 + l1 (t))−1 (δ + 500). So there are two poles, one of which is δ = −500 being the invariant zero of the system Σ(C(t); A(t); F(t)). As term (cos(t) + 1:2 + l1 (t))−1 is time varying, it cannot be commuted with the term (δ + 0:2l2 (t)). Because only right factors of polynomials in LTV cases are roots of polynomials, it is difficult to compute poles. If l1 (t) = −cos(t), the polynomial becomes 2(δ + 0:2l2 (t))(δ + 500). Let l2 (t) = 5, the observer has two negative poles. The estimation errors of two state variable with an initial condition x(0) = [2 1]T are displayed in Fig. 6. The estimation of d is dˆ = 5(y′ −[ 0:4(cos(t) + 1:2) −0:008 ] x). ˆ

The designed observer is stable and rebuilds correctly real states and the unknown input which is a time function. In the LTV case, the proposed observer (3) is still valid but matrices in (3) must be redesigned, for example with symbolic calculation. The second case (with detector on I : L) is more complex, because two time varying poles must be paced (not studied here). Finally, with the known inputs which are rarely kept on in cited works, the observer (3) is a good extension of the observer in Daafouz et al. (2006), but derivations of the control input are necessary. The flatness approach can be a good solution, if the output to be controlled is flat, which is the case in the above example, whatever the position of the sensor. 5. CONCLUSION In this paper, input and state observers are proposed for linear bond graph models. The analysis of the solvability of the UOI problem is proposed with a graphical procedure, and an extension is proposed for the synthesis of observers with control inputs. The classical assumption on the relative degree between the unknown input and the output variable is not necessary in this context. For stability requirement, some invariant zeros must be stable. It is shown that the algebraic approach is necessary for the study of invariant zeros in the LTV case. A direct extension is the control of systems modelled by bond graph in which the output variable is not the measured output,

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