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Mathematical and Computer Modelling of Dynamical Systems, 2015 Vol. 21, No. 1, 38–57, http://dx.doi.org/10.1080/13873954.2013.874360 A procedure to linearize a class of non-linear systems modelled by bond graphs a b Gilberto Gonzalez Avalos * and Rene Galindo Orozco a b Faculty of Electrical Engineering, University of Michoacan, Morelia, Mexico; Department of Electrical Engineering, University of Nuevo Leon, San Nicolas de lo Garza, Mexico (Received 11 April 2012; final version received 9 December 2013) A procedure is proposed to obtain the linearization of a class of non-linear physical systems using bond graphs. Also, a junction structure of a non-linear bond graph considering linearly dependent and independent state variables is described. From the junction structure of the non-linear bond graph a procedure to build a linearized bond graph is presented. Finally, an example of a Programmable Universal Manipulation Arm (PUMA) manipulator is given. Keywords: bond graph; non-linear systems; linearization; DC motor 1. Introduction Although almost every physical system contains non-linearities, oftentimes its behaviour is within a certain operating range of an equilibrium point which can be reasonably approximated by that of a linear model [1]. One reason for approximating the non-linear system by a linear model is that, by doing so, one can apply rather simple and systematic linear control design techniques [1]. A linearized system is useful in order to know the behaviour of the system when it is perturbed, such that the new and old equilibrium points are nearly equal. The system equations are linearized around the operating points. The new linear equations, thus derived, are assumed to be valid in a region near the equilibrium point [2]. There are several applications of linearized systems; for example, small-signal stability [2], which corresponds to the ability of the power system to maintain synchronism when it is subjected to small disturbances, i.e., the property of a power system that enables it to retain a state of operating conditions and to regain an acceptable state of equilibrium after being subjected to a disturbance. In this context, a disturbance is considered to be small if the equations that describe the response of the system can be linearized for the purpose of analysis [3]. There are many references and applications of linearized systems [1,4–10]. The following papers can be cited: in [11], it is shown how the well-known linearization technique can be used to solve the problem of fault diagnosis, for non-linear systems are subject to actuator failures, but this method does not show how the fault diagnosis can be performed directly in a non-linear system. In [12], the authors propose a method to linearize and simplify the manipulator dynamics, by examining the complete expanded Lagrange equation in order to redesign the link’s inertia property; as a result, the *Corresponding author. Email: gilmichga@yahoo.com.mx © 2014 Taylor & Francis Mathematical and Computer Modelling of Dynamical Systems 39 coefficients of all (or nearly all) the non-linear terms become zero; this is a well-known technique in robotics since the 1980s. Also, an equivalent linearization and the averaging methods are employed to simplify the hysteretic non-linear model of a single degree of freedom suspension in [13]. A bond graph is a useful and important tool for physical system modelling. This is based on power representation, and it enables the description of the system through energy storage and dissipative elements [14]. A bond graph can represent a variety of energy types, whose junction structure can give valuable information of the properties of the physical system [14,15]. An important property of the bond graph theory is the causal path, which allows determination of system structural properties such as observability and controllability [16], or the relation between state variables in order to linearize a class of non-linear system, which can be represented by a bond graph. In [17], the power and energy analysis in linearized physical systems in a bond graph approach is proposed. Karnopp states that each non-linear element can be replaced by a linearized element in a bond graph model. This paper permits consideration of the complete non-linear bond graph with linearly dependent and independent state variables and non-linear resistors giving a Lemma and a procedure to build a linearized bond graph from the non-linear bond graph. A class of non-linear physical systems characterized by functions of state variables fxðÞ are mod- elled and this paper proposes to linearize this type on non-linear system; for example, synchronous machines [9] and robotic manipulators [18]. Undoubtedly, the traditional approach to linearize non-linear systems from their state space representation can be used. However, the mathematical model has to be obtained. Hence, this paper permits linking the analysis of linear systems to a class of non-linear systems when these systems are modelled and linearized in a bond graph approach. Section 2 summarizes the algebraic linearization of non-linear systems using the Taylor series expansion around the equilibrium point. Section 3 describes the bond graph model of a class of non-linear physical systems. A Lemma and procedure to obtain the linearization from the non-linear bond graph are proposed in Section 4. An example of a Programmable Universal Manipulation Arm (PUMA) manipulator illustrates the pro- posed methodology in Section 5. Finally, Section 6 presents the conclusions. 2. Linearization Non-linear models with small disturbances around some equilibrium point act virtually in the same manner as their simpler linearized models predict, in some cases. A linear state equation [19] is useful as an approximation of a non-linear state equation, xt _ðÞ ¼ fxðÞ ðÞt ;utðÞ xtðÞ¼ x (1) 0 0 n p where the state xtðÞ 2 < is the state and the input utðÞ 2 < is the input. Let ðÞ 1 be solved for a particular input signal called nominal input ~ uðtÞ and a particular initial state called nominal initial state ~x to obtain a unique nominal solution, often called a nominal trajectory ~xtðÞ [19], i.e., ~xtðÞ ¼ fðÞ ~xtðÞ; ~utðÞ and ~xtðÞ¼ ~x , 0 0 dt where ~xtðÞ has to be reachable by the non-linear model. It is interesting to know the behaviour of the non-linear state equation for an input and state that are close to the nominal values. That is, consider, 40 G. Gonzalez Avalos and R. Galindo Orozco utðÞ ¼ ~utðÞ þ u ðÞ t (2) x ¼ ~x þ x (3) 0 0 0δ wherekk x andkk u are appropriately small for t t . 0δ δ 0 2 2 Assume that the corresponding solution remains close to ~xtðÞ at each t; and let us write xtðÞ ¼ ~xtðÞ þ x ðÞ t (4) where x ðÞ t is the distance to the nominal trajectory. Substituting ðÞ 2 and ðÞ 4 into ðÞ 1 , and assuming that ðÞ 1 is continuous and differenti- able, expand the right side using Taylor series around ~xtðÞ and ~utðÞ; and then retain only the terms of order one [19], getting, d d @f @f ~xtðÞ þ x ðÞ t ﬃ fðÞ ~xtðÞ; ~utðÞþ x ðÞ t þ u ðÞ t (5) δ δ δ dt dt @x @u ðÞ ~xtðÞ;~utðÞ ðÞ ~xtðÞ;~utðÞ @f @f @f @f i i where and are the Jacobians, matrices with i; j entry and . @x @u @x @u j j ðÞ ~xtðÞ;~utðÞ ðÞ ~xtðÞ;~utðÞ Since ~xtðÞ ¼ fðÞ ~xtðÞ; ~utðÞ , ~xtðÞ¼ ~x ; the relation between x ðÞ t and u ðÞ t is approxi- 0 0 δ δ dt mately described by a time-invariant linear state equation of the form x _ ðÞ t ¼ A x ðÞ t þ B u ðÞ t (6) δ δ δ δ δ where A and B are the matrices of partial derivatives evaluated on the nominal trajectory, δ δ that is, @f A ¼ (7) @x ðÞ ~xtðÞ;~utðÞ @f B ¼ (8) @u ðÞ ~xtðÞ;~utðÞ In this development, a nominal solution is assumed to exist for all t t so that the linearization makes sense as an approximation. The search for equilibrium points is only the starting point before linearizing a model. Using the bond graph representation of a system, Breedveld [20] has proposed a proce- dure to determine the equilibrium state where energy stored in each storage element is constant, thus the efforts (respectively, the flow) of the I elements (respectively, the C elements) have then to be zero, and Breedveld proposes replacing the I elements by null effort sources and the C elements by null flow sources. In [21], Bideaux et al., bicausality concepts are used to solve the problem of the existence and determination of the equilibrium set of a system using a bond graph. Mathematical and Computer Modelling of Dynamical Systems 41 3. Modelling in bond graph Consider the scheme of a multiport non-linear system represented by a bond graph in a preferential integral causality assignment which has the key vectors of Figure 1 [22]. In Figure 1, a bond graph is represented by MS ; MS ,ðÞ I; C andðÞ R , which denote e f the sources, and the storage and dissipation fields, respectively, and (0, 1, MTF, MGY), which denotes the junction structure. Note that modulated transformers, MTF, and modu- lated gyrators, MGY, can be modulated by state functions or input signals. n m The states xtðÞ 2 < and x ðÞ t 2 < are composed of energy variablesðÞ ptðÞ and qtðÞ associated with I and C elements in integral ðÞ xtðÞ and derivativeðÞ x ðÞ t causality p n m assignments, respectively, utðÞ 2 < denotes the plant input, ztðÞ 2 < and z ðÞ t 2 < are the co-energy state variables in integralðÞ ztðÞ and derivativeðÞ z ðÞ t causality assign- r r ments, respectively, and D ðÞ t 2 < and D ðÞ t 2 < are a mixture of efforts etðÞ and in out flows ftðÞ showing the energy exchanges between the dissipation field and junction structure. A class of non-linear systems can be obtained by applying active bonds of state functions to modulate the MTF and/or MGY. The dissipation field is divided into two l nl parts: linear resistors, D ðÞ t and non-linear resistors D ðÞ t . in in Most physical systems are non-linear, such as robotics, induction and synchronous machines, and many systems with non-linear resistors can be modelled by bond graphs with a junction structure defined as follows: 2 3 ztðÞ 2 3 2 3 11 12 xt _ðÞ S ðÞ x S ðÞ x S ðÞ x S ðÞ x S ðÞ x 11 13 14 6 7 12 12 6 D ðÞ t 7 6 7 6 7 out l 11 11 11 6 7 D ðÞ t S ðÞ x S ðÞ x 0 S ðÞ x 0 6 7 6 7 in 21 22 23 6 nl 7 6 7 ¼ 6 7 (9) D ðÞ t 6 out 7 nl 21 21 4 5 4 5 D ðÞ t S ðÞ x 00 S ðÞ x 0 6 7 in 21 23 4 utðÞ 5 z ðÞ t S ðÞ x 000 0 d 31 x _ ðÞ t The constitutive relations of the elements are: ztðÞ ¼ FxðÞ t (10) z ðÞ t ¼ F x ðÞ t (11) d d d l l D ðÞ t ¼ LD ðÞ t (12) out in Source field (MS , Ms ) e f u(t) z (t) x (t) d D (t) Dissipation Storage in Junction Structure field field x(t) (0, 1, MTF, MGY) (R) (C, I) z(t) D (t) out f(x(t)) Figure 1. Key vectors of a non-linear bond graph. 42 G. Gonzalez Avalos and R. Galindo Orozco nl nl D ðÞ t ¼ f D ðÞ t (13) out in and the state equation is ExðÞxt _ðÞ ¼ AxðÞ ; x _ xtðÞ þBxðÞutðÞ þHxðÞ ; u (14) where ExðÞ ¼ I S ðÞ x F S ðÞ x F (15) 14 31 11 11 1 Ax; ¼ S ðÞ x þ S ðÞ x MxðÞS ðÞ x þ S ðÞ x F S ðÞ x F (16) 11 14 31 12 21 d 11 11 BxðÞ ¼ S ðÞ x þ S ðÞ x MxðÞS ðÞ x (17) 12 23 12 21 21 HxðÞ ; u ¼ S ðÞ x f S ðÞ x FxðÞ t þ S ðÞ x utðÞ (18) 12 L 21 23 being MxðÞ ¼ LI S ðÞ x L (19) FromðÞ 14 , _ ðÞ x½ AxðÞ ; x _ xtðÞ þBxðÞutðÞ þHxðÞ ; u (20) xtðÞ ¼ E and replacing xðÞ t into Ax; x inðÞ 20 , EquationðÞ 14 can be reduced to xt _ðÞ ¼ AxðÞxtðÞ þBxðÞutðÞ (21) where AxðÞ and BxðÞ are matrices obtained by solving xt _ðÞ inðÞ 20 . EquationðÞ 21 is obtained by solving the system of differential equations that result from substituting ðÞ t intoðÞ 20 . A graphical procedure to obtain a linearized bond graph is proposed in Section 4. 4. A bond graph approach to linearize a class of non-linear systems Bond graph represents a physical structure denoting power exchange in the physical system. It is possible to code on the graph its mathematical structure, and then a structure showing the causal relationships among the signals on the system can be obtained. In order to linearize a non-linear system modelled by a bond graph, the following Lemma allows us to obtain additional terms that define the new causal paths to construct a bond graph of the linearized system. Lemma. Consider a state equation of a class of non-linear systems modelled by bond graphs of the form Mathematical and Computer Modelling of Dynamical Systems 43 ExðÞxt _ðÞ ¼ AxðÞ ; x _ xtðÞ þBxðÞutðÞ þHxðÞ ; u (22) where AðÞ x; x _ ;BxðÞ and EðÞ x are matrices that depend on states and HðÞ x; u is a non- linear function of states and inputs determined by a junction structure matrix defined in ðÞ 9 ; and a linear constitutive relation of the storage field and a non-linear constitutive relation of the dissipation field. Then, a bond graph of the linearized system is described as follows: x _ ðÞ t ¼ A x ðÞ t þ B u ðÞ t (23) δ δ δ δ δ where ExðÞj A ¼ AxðÞ ; x _j þ A ðÞ ~x þ A ðÞ ~x þ A ðÞ ~x (24) δ x u h ðÞ ~xtðÞ;~utðÞ ðÞ ~xtðÞ;~utðÞ ExðÞj B ¼ BxðÞj þ B ðÞ ~x þ B ðÞ ~x (25) δ u h ðÞ ~xtðÞ;~utðÞ ðÞ ~xtðÞ;~utðÞ being @AxðÞ ; x _ @E ðÞ x A ðÞ ~x ¼ þExðÞ AxðÞ ; x _ F ~ztðÞ (26) @x @x ðÞ ~xtðÞ;~utðÞ @BxðÞ @E ðÞ x A ðÞ ~x ¼ þExðÞ BxðÞ ~utðÞ (27) @x @x ðÞ ~xtðÞ;~utðÞ @HxðÞ ; u @E ðÞ x A ðÞ ~x ¼ þExðÞ HxðÞ ; u (28) @x @x ðÞ ~xtðÞ;~utðÞ hi @ S ðÞ x F S ðÞ x B ðÞ ~x ¼ ~ztðÞ (29) @u ðÞ ~xtðÞ;~utðÞ @HxðÞ ; u B ðÞ ~x ¼ (30) @u ðÞ ~xtðÞ;~utðÞ with 11 11 @ S ðÞ x MxðÞS ðÞ x @AxðÞ @S ðÞ x 12 21 F ¼ þ þ @x @x @x hi @ S ðÞ x F ðÞ x d 31 (31) @x 11 11 @BxðÞ @S ðÞ x @ S ðÞ x MxðÞS ðÞ x 12 23 ¼ þ (32) @x @x @x 12 nl @HxðÞ ; u @S ðÞ x @f D L in 12 nl 12 ¼ f D þ S ðÞ x (33) in 12 @x @x @x 44 G. Gonzalez Avalos and R. Galindo Orozco it can be built from the original non-linear bond graph modulating MTF and/or MGY by nominal trajectory, ~x; the non-linear resistors are linearized and add new causal paths. Moreover, the new causal paths correspond to A ðÞ ~x ; A ðÞ ~x ; A ðÞ ~x ; B ðÞ ~x and B ðÞ ~x . The x u h u h A ðÞ ~x causal paths begin at the storage elementthat is changed by a source ~ztðÞ until it reaches another storage element x ðÞ t .The A ðÞ ~x causal paths begin at the correspond- δ u ing source ~utðÞ and the paths finishe at the corresponding storage element x ðÞ t . B ðÞ ~x δ u are the linearized causal paths between storage elements in derivative and integral causality assignments; and A ðÞ ~x and B ðÞ ~x are the new linearized causal paths, due to h h causal paths between states, inputs and non-linear resistor. The proof is presented in Appendix. In this paper, the identification of the causal paths with MTF or MGY modulated by a function of state variables and/or non-linear resistors is the fundamental part to obtain the linearization of the system modelled by a bond graph, because these non-linear causal paths describe the non-linear terms of the state equations. A procedure to construct the linearized bond graph from a non-linear bond graph of a physical system is presented. 4.1. Procedure (1) Given a non-linear bond graph, the non-linear causal paths and/or non-linear causal loops have to be identified. (a) The non-linear causal paths begin atðÞ I ; C orðÞ Se ; Sf element, through i i i i eitherðÞ MTF; MGY modulated by a function of state variables and arrive to I ; C element. j j (b) The non-linear causal loops begin and finish at anðÞ I ; C element and i i ðÞ MTF; MGY modulated by a function of state variables. (2) The linearization is in the vicinity of the known nominal trajectory, then MTF and/or MGY modulus of step 1 will change by TF and/or GY modulated by the nominal trajectory. (3) Each causal path of step 1 will include a new causal path. Considering wxðÞ ¼½ ðÞ wxðÞ ðÞ wxðÞ ðÞ wxðÞ , the Jacobian of ðÞ wxðÞ is 1 i n i @ðÞ wxðÞ @ðÞ wxðÞ i i i and ¼ G is the function obtained by the Jacobian with the j;m @x @x position row j and column m. Note that G is the partial derivative function of j;m the non-linear causal path gain. (a) The new causal paths due to non-linear causal paths between storage ele- ments are shown in Figure 2, where wxðÞ ¼ E ðÞ x S ðÞ x . (b) The new causal paths due to non-linear causal paths between sources and storage elements are shown in Figure 3, where 11 11 wxðÞ ¼ E ðÞ x ½S ðÞ x þ 13 S ðÞ x MxðÞS ðÞ x : 12 23 (c) The new causal paths due to non-linear causal loops are shown in Figure 2, 1 11 11 1 1 where wxðÞ ¼ E ðÞ x S ðÞ x MxðÞS ðÞ x or wxðÞ ¼ E ðÞ x S ðÞ x F S ðÞ x . 14 31 12 21 d Mathematical and Computer Modelling of Dynamical Systems 45 Non-linear causal paths and loops Additional causal paths I:L (a) x (t) (a) C:C δ i f(x(t)) i G (x) J,k ~ ~ [x (t)] [x (t)] [x (t)] i (x(t),u(t)) j δ 0 1 MTF 1 I:L 1 MTF Se:e (t) TF i j (b) (b) C:Ci C:C x (t) f(x(t)) j δ G (x) J,k ~ ~ [x (t)] [x (t)] (x(t),u(t)) [x (t)] i j δ 0 1 Se:e (t) MGY MGY 0 TF 0 C:C (c) I:Li C:C (c) x (t) f(x(t)) j G (x) J,k [x (t)] ~ [x (t)] [x (t)] (x(t),u(t)) j δ 1 MTF 0 Sf:f (t) i TF MTF 0 C:C (d) I:L I:Li j (d) x (t) f(x(t)) δ G (x) J,k ~ ~ [x (t)] [x (t)] [x (t)] (x(t),u(t)) j j 1 0 1 Sf:f (t) MGY 1 I:L MGY i TF Figure 2. Cases to include additional causal paths. Additional causal paths Non-linear causal paths (a) I:L (a) x (t) f(x(t)) δ G (x) [x (t)] [x (t)] J,k ~ ~ j δ (x(t),u(x)) j MSe:e (t) 1 I:L i 0 MTF Se:e (t) TF 1 i MTF j x (t) C:C (b) (b) i δ j k f(x(t)) G (x) J,k [x (t)] ~ ~ [x (t)] δ j j (x(t),u(x)) MSe:e (t) Se:e (t) TF 0 MGY C:C i 0 MGY i 0 j (c) C:C (c) x (t) j i δ f(x(t)) k G (x) J,k [x (t)] ~ δ [x (t)] j (x(t),u(x)) Sf:f (t) 0 MSf:f (t) 1 MTF TF C:C i i MTF 0 j I:L (d) x (t) (d) f(x(t)) δ i k G (x) [x (t)] J,k δ [x (t)] ~ ~ (x(t),u(x)) MSf:f (t) 1 Sf:f (t) I:L i 1 MGY TF 1 i MGY j 1 11 11 Figure 3. New causal paths for E ðÞ x S ðÞ x þ S ðÞ x MxðÞS ðÞ x . 12 23 (d) The new causal paths due to non-linear causal paths or loops with non-linear resistors are shown in Figures 4 and 5, where wxðÞ ¼ E ðÞ x HxðÞ. FromðÞ 13 andðÞ 18 , nl @HxðÞ ; u @f D ðÞ t @S ðÞ x 12 in 12 nl ¼ S ðÞ x þ D ðÞ t 12 out @x @x @x nl @f D ðÞ t L in the component S ðÞ x gives new causal paths which are shown in Figure 4. @x @S ðÞ x nl Figure 5 determines new causal paths due to the component D ðÞ t . In this case, out @x nl the non-linear resistors R : f D are changed by in k 46 G. Gonzalez Avalos and R. Galindo Orozco Additional causal paths Non-linear causal paths R:φ (e ) j j x (t) C:C f(x (t)) (a) k δ i i G (x) J,k ~ ~ [x (t)] [x (t)] (x(t),u(t)) j 0 1 1 Se:z (t) MGY C:C MTF TF 1 i j (a) R:φ (f ) (b) I:L j j x (t) f(x (t)) i δ i k G (x) J,k [x (t)] ~ ~ [x (t)] j (x(t),u(t)) Sf:z (t) MGY 1 I:L 1 MTF 0 TF (b) 1 i j Figure 4. Non-linear causal loops with non-linear resistors. Non-linear causal loops Additional causal paths R:φ (e ) j j ~ NL f(x (t)) C:C i x (t) k Sf:Dout i δ G (x) J,k [x (t)] ~ ~ [x (t)] (x(t),u(x)) i 1 MTF 1 0 1 TF MTF 0 (a) (a) C:C NL R:φ (f ) j j C:C Se:Dout i f(x (t)) j k G (x) J,k [x (t)] (x(t),u(x)) δ [x (t)] i 1 0 1 TF MGY 1 MGY C:C (b) (b) R:φ (f ) x (t) j j ~ NL I:L k f(x (t)) Sf:Dout i k j G (x) J,k ~ ~ [x (t)] (x(t),u(x)) [x (t)] TF MGY 1 1 I:L 1 0 MGY 1 i (c) (c) R:φ (f ) ~ NL x (t) j j I:L Se:Dout i δ f(x (t)) i k j G (x) k J,k [x (t)] ~ ~ (x(t),u(x)) [x (t)] (d) TF MTF 1 I:L 1 MTF (d) 0 1 i Figure 5. Non-linear causal loops with non-linear resistors. R : R j (34) LIN ðÞ ~xtðÞ;~utðÞ nl @f D ðÞ L in 21 k 1 where R ¼ S ðÞ x F with LIN k @x 2 3 nl @f D ðÞ L in k 1 F 0 @x nl 6 1 7 @f D L in 6 7 k 1 . . . F ¼ (35) 6 . . . 7 . . . @x 4 5 nl @fðÞ D in k 1 0 F @x and k is the index of the non-linear resistor of the corresponding non-linear causal loop. Also, R can be written by LIN hi nl nl @f D @f D ðÞ ðÞ L L 21 in 1 21 in 1 k k R ¼ (36) LIN S ðÞ x F S ðÞ x F 21 @x 1 21 @x n 1k 1 nk n Mathematical and Computer Modelling of Dynamical Systems 47 in a compact form is nl @f D 21 in 1 R ¼ S ðÞ x F (37) LIN 21 1;;n 1k;;nk @x 1;;n (e) The new causal paths due to non-linear causal paths with non-linear resistors beginning at sources to storage elements are shown in Figure 6, where @ðÞ wxðÞ @ðÞ wxðÞ i 1 i i wxðÞ ¼ E ðÞ x HxðÞ andðÞ wxðÞ is and ¼ G . i j;m @u @u Figure 2 shows some examples of the new causal paths obtained due, to the linearization 1 1 11 11 1 1 of the terms E ðÞ x S ðÞ x ; E ðÞ x S ðÞ x MxðÞS ðÞ x and E ðÞ x S ðÞ x F S ðÞ x in the 11 14 31 12 21 d corresponding non-linear causal paths. This figure indicates that the storage element is changed into a source arriving to the other storage element and the TF modules is given by the corresponding partial derivative. 1 11 11 The new causal paths for the linearization of E ðÞ x S ðÞ x þ S ðÞ x MxðÞS ðÞ x are 12 23 shown in Figure 3 and they are similar with respect to Figure 2, but the first storage element is now a source on the non-linear bond graph. The linearization due to non-linear causal loops with non-linear resistors gives new nl @f D ðÞ t ðÞ L in causal paths, which are shown in Figure 4 for S ðÞ x and Figure 5 for @x @S ðÞ x nl D ðÞ t . Figure 4 shows the non-linear causal paths with non-linear resistors and @x out is equivalent to Figure 2 considering the appropriate partial derivative. Another part of the linearization formed by the non-linear causal paths with non- linear resistors, is to replace the non-linear resistor by a source corresponding to the causality assignment of the resistor in the non-linear bond graph which is shown in Figure 5. Non-linear causal paths Additional causal paths x (t) R:φ (e ) C:C (a) Mse:u (t) k m m j i f(x (t)) i G (x) J,k [x (t)] ~ ~ [x (t)] δ j (x(t),u(x)) 0 1 0 TF MGY 0 C:C MTF Se:u (t) (a) Msf:u (t) R:φ (e ) C:C x (t) i m m j δ f(x (t)) k (b) k G (x) J,k [x (t)] [x (t)] ~ ~ (x(t),u(x)) 0 ~ MGY 0 Sf:u (t) TF MTF C:C (b) 0 j Msf:u (t) R:φ (e ) I:L x (t) i m m j δ f(x (t)) k k (c) G (x) [x (t)] J,k ~ ~ [x (t)] j δ (x(t),u(x)) 0 ~ 1 MTF 1 TF Sf:u (t) MGY 1 I:L (c) i x (t) Mse:u (t) R:φ (e ) δ I:L i m m j k f(x (t)) (d) i G (x) J,k ~ ~ [x (t)] (x(t),u(x)) j [x (t)] 0 1 1 TF MTF 1 I:L MGY Se:u (t) (d) Figure 6. Non-linear causal paths with non-linear resistors from sources to storage elements. 48 G. Gonzalez Avalos and R. Galindo Orozco Figure 7. General structure of a linearized bond graph. Figure 6 shows the linearization of the non-linear causal paths beginning at sources with non-linear resistors and ending at a storage element, which gives new causal pathways. A linearized model describes the behaviour of perturbations, or changes in the state variables relative to some nominal condition. Hence, a general structure from a linearized bond graph, using the proposed procedure, is shown in Figure 7. Note that, Figure 7 represents the obtained linearized bond graph and its junction structure. The TF and GY modules are the nominal trajectory of the original non-linear bond graph, fðÞ ~xtðÞ . Also, additional bonds according to the causal paths that represent non-linear parts of the original bond graph model, with S and S sources, vector ~utðÞ; are e f included. Finally, non-linear resistors that are a part of the dissipation field are linearized. Thus, state variables representation ðÞ 23 using junction structure or causal paths of the linearized bond graph can be obtained. In Section 5, an example, applying the proposed Lemma and procedure, is solved. 5. Example A simple two-degrees of freedom (DOF) manipulator, but three-dimensional mode, appears in Figure 8. This can be regarded as a simplified PUMA with the elbow and wrist locked at appropriate angles and zero joint offset [18]. The second link, although moving in three dimensions, rotates around a fixed point: joint 2. Its dynamics are, therefore, determined by an Euler ring. The first link is a simple one-dimensional rotating inertia coupled to the second link by a joint. The angular velocities of the second link around the × and y axes w and w are entirely determined x y by that of the first link w , w ¼ w sinðÞ θ (38) x 1 2 w ¼ w cosðÞ θ (39) y 1 2 The non-linear bond graph in a preferential integral causality assignment of the PUMA manipulator is shown in Figure 9. Also, consider that the system has two Mathematical and Computer Modelling of Dynamical Systems 49 Joint 2 Link 1 Joint 1 Figure 8. Scheme of 2 DOF PUMA. R:R l:i R:R l:i 21 24 l:i R:φ (f ) z 28 1 20 22 MSe:S 25 2 1 1 MSe:S 19 23 MTF:cos(q ) C:C l:i C:C 17 1 MGY:p R:φ (f ) x 27 MGY:p 14 15 1 1 l:i MGY l:i 8 26 R:φ (f ) y 26 MTF:sin(q ) Figure 9. Bond graph of the PUMA manipulator. linear resistorsðÞ R : R ; R : R and three non-linear resistors ðR : fðÞ f ; 1 2 27 R : fðÞ f ;R : fðÞ f Þ. 26 28 y z The key vectors for this bond graph are: 2 3 2 3 2 3 2 3 p e f 2 2 2 6 7 6 7 6 7 p e f 24 24 24 nl 6 7 6 7 6 7 4 5 xtðÞ ¼ ; xt _ðÞ ¼ ; ztðÞ ¼ ; D ðÞ t ¼ f 4 5 4 5 4 5 in q f e 3 3 3 q f e 23 23 23 2 3 2 3 2 3 2 3 p e f 8 8 8 6 7 6 7 6 7 p e f 12 12 12 nl 6 7 6 7 6 7 4 5 x ðÞ t ¼ ; x _ ðÞ t ¼ ; z ðÞ t ¼ ; D ðÞ t ¼ e d d d 27 4 5 4 5 4 5 out p e f 13 13 13 p e f 19 19 19 Link 2 50 G. Gonzalez Avalos and R. Galindo Orozco f e e l 6 l 6 1 D ðÞ t ¼ ; D ðÞ t ¼ ;utðÞ ¼ in out f e e 21 21 25 the constitutive relations are 1 1 1 1 F ¼ diag ; ; ; (40) i i C C 2 1 2 1 1 1 1 1 F ¼ diag ; ; ; (41) i i i i z x y L ¼ diagfg R ; R (42) 2 1 e ¼ ’ðÞ f ; e ¼ ’ðÞ f ; e ¼ ’ðÞ f (43) 26 y 26 27 x 27 28 z 28 and the junction structure is 2 3 2 3 2 3 6 7 e 0 h 10 10 0 0 11 0 10 0 0 f 2 1 24 6 7 6 7 6 76 7 e h 00 10 1 cosðÞ q sinðÞ q 00 1 0 sinðÞ q cosðÞ q 1 e 24 1 3 3 3 3 3 6 7 6 76 7 6 7 6 76 7 f 1 0 0 0 0 0 00 0 00 0 00 0 e 3 23 6 7 6 76 7 6 7 6 76 7 f 0 1 0 0 0 0 00 0 00 0 00 0 e 23 6 6 7 6 76 7 6 7 6 76 7 f 1 0 0 0 0 0 00 0 00 0 00 0 e 6 21 6 7 6 76 7 6 7 6 76 7 f 0 1 0 0 0 0 00 0 00 0 00 0 e 21 26 6 7 6 76 7 6 7 6 76 7 f ¼ 0 cosðÞ q 0 0 0 0 00 0 00 0 00 0 e 26 3 27 6 7 6 76 7 6 7 6 76 7 f 0 sinðÞ q 0 0 0 0 00 0 00 0 00 0 e 27 3 28 6 7 6 76 7 6 7 6 76 7 f 1 0 0 0 0 0 00 0 00 0 00 0 e 28 1 6 7 6 76 7 6 7 6 76 7 f 1 0 0 0 0 0 00 0 00 0 00 0 e 8 25 6 7 6 76 7 6 7 6 76 7 f 0 sinðÞ q 0 0 0 0 00 0 00 0 00 0 e 12 3 8 6 7 6 76 7 4 5 4 56 7 f 0 cosðÞ q 0 0 0 0 00 0 00 0 00 0 e 13 3 12 6 7 4 5 f 0 1 0 0 0 0 00 0 00 0 00 0 e 19 13 (44) where h ¼ p sinðÞ q p cosðÞ q . 1 13 3 12 3 By substitutingðÞ 40 ;ðÞ 41 ;ðÞ 42 andðÞ 44 intoðÞ 15 ;ðÞ 16 ,1ðÞ7 ;ðÞ 18 andðÞ 14 the state equation is given by 2 3 2 3 2 3 2 3 2 3 R h 1 2 1 p p 2 i i C p e 2 1 2 2 f 1 6 7 6 h R 7 2 1 1 1 6 7 6 7 6 7 p p e 6 7 6 7 24 g 25 24 i i C 2 1 1 6 7 6 a 7 6 7 E ¼ þ þ (45) 6 7 6 7 4 5 4 5 4 5 q 0 4 5 4 00 0 5 q 3 0 q 0 23 0 0 00 23 1 no i i þi z x where i ¼ i , E ¼ diag 1 þ ; 1 þ ; 1; 1 and g ¼ g þ g being g ¼ x y a 1 2 1 i i 2 1 p cosðÞ q p sinðÞ q 24 3 24 3 cosðÞ q f and g ¼sinðÞ q f . 3 2 3 y x i i 1 1 It can be seen that ExðÞ ¼ E is a constant matrix with i ¼ i . S ðÞ x is also a constant x y 13 matrix. Then, fromðÞ 23 toðÞ 27 the proposed Lemma is reduced to @AðÞ ~x @HðÞ ~x; ~ u Ex _ ðÞ t ¼ AðÞ ~x þ F ~ztðÞ þ x ðÞ t þ BðÞ ~x u ðÞ t δ δ δ @x @x Mathematical and Computer Modelling of Dynamical Systems 51 where AðÞ ~x and BðÞ ~x are the matrices of non-linear system evaluated on the nominal @AðÞ ~x @S ðÞ ~x trajectory. Then, F ~ztðÞ ¼ is obtained by the linearization of the non- @x @x linear causal paths. Now, the proposed procedure to construct a linearized bond graph is applied. The nominal trajectory is defined by ~x; ~z; ~z and ~ u: By identifying the non-linear parts of Figure 9, the causal paths ðÞ 7! and causal loops ðÞ $ between energy storage ele- ments are, 2 ! 4 ! 5 ! 7 ! 10 ! 16 ! 18 ! 22 ! 24 I : i 7!I : i 2 1 2 ! 4 ! 5 ! 9 ! 11 ! 17 ! 20 ! 22 ! 24 The first column of S ðÞ x is given by ðÞ 44 and the Jacobian ofðÞ S ðÞ x is 11 11 @ðÞ S ðÞ x ½ 11 1 @½ p sinðÞ q p cosðÞ q 1 2 13 3 12 3 G ¼ ¼ ¼ p cosðÞ q þ p sinðÞ q ¼ h then the new cau- 13 3 12 3 2 2;3 @x @q 3 3 sal paths, by using Figure 2(d), are shown in Figure 10. Other non-linear causal paths are described by 24 ! 22 ! 18 ! 16 ! 10 ! 7 ! 5 ! 4 ! 2 I : i 7!I : i 1 2 24 ! 22 ! 20 ! 17 ! 11 ! 9 ! 5 ! 4 ! 2 and the new causal paths are shown in Figure 11. The non-linear causal paths and loops with non-linear resistors of Figure 9 are: 2452828542gI : i $ R : fðÞ f 2 28 24222017262617202224gI : i $ R : fðÞ f 1 26 24221816272716182224gI : i $ R : fðÞ f 1 27 @HxðÞ ;u As ExðÞ ¼ E thenðÞ 28 is reduced to A ðÞ ~x ¼ . The Jacobian due to non- @x ðÞ ~xtðÞ;~utðÞ linear resistor is: TF Sf:f (t) MGY 1 l:i 2 1 ~~ ~~ p cos (q ) + p sin (q ) 13 3 12 3 Figure 10. New causal paths for I : i 7! I : i . 2 1 TF Sf:f (t) MGY 1 l:i 24 2 ~ ~ ~ –p cos (q ) – p sin (q ) 13 3 12 3 Figure 11. New causal paths for I : i 7! I : i . 1 2 52 G. Gonzalez Avalos and R. Galindo Orozco 2 3 @fðÞ p =i 2 2 00 0 @p 6 7 @g @g @HxðÞ ; u a a 6 7 0 0 ¼ @p @ 6 24 q 7 @x 4 5 00 0 0 00 0 0 @g where ¼ g ðÞ x þ g ðÞ x being g ðÞ x ¼ sinðÞ q e cosðÞ q e and g ðÞ x ¼ 3 4 3 3 26 3 27 4 @q i @e i @e 1 26 1 27 cosðÞ q sinðÞ q . 3 3 p @q p @q 24 3 24 3 @fðÞ p =i 2 2 @g z a The bond graphs of the partial derivatives and are @p @p 2 24 ðÞ ~xtðÞ;~utðÞ ðÞ ~xtðÞ;~utðÞ obtained by causal loops between the linearized resistors and the corresponding storage elements withðÞ 34 . The new causal paths due to g ðÞ x ; by using Figure 4(b), are shown in Figure 12. Figure 13 shows the new causal paths for g ðÞ x by using Figure 5(d). By connecting bond graphs of the new causal paths to the non-linear bond graph under the nominal trajectory, the linearized bond graph of the manipulator is shown in Figure 14. The linearized bond graph represents the linearization of the manipulator by using the junction structure matrix given by 2 3 2 3 2 3 ~ ~ 6 7 e 0 h 1 þ h 0 10 0 0 11 0 10 0 0 f 2 1 3 24 6 7 6 7 6 ~ ~ 76 7 e h 0 h 10 1 cosðÞ ~ q sinðÞ ~ q 00 10 sinðÞ ~ q cosðÞ ~ q 1 e 24 1 4 3 3 3 3 3 6 7 6 76 7 6 7 6 76 7 f e 3 1 0 0 0 0 0 0 0 00 00 0 0 0 23 6 7 6 76 7 6 7 6 76 7 f e 23 0 1 0 0 0 0 0 0 00 00 0 0 0 6 6 7 6 76 7 6 7 6 76 7 f e 6 1 0 0 0 0 0 0 0 00 00 0 0 0 21 6 7 6 76 7 6 7 6 76 7 f e 21 0 1 0 0 0 0 0 0 00 00 0 0 0 26 6 7 6 76 7 6 7 6 76 7 f 0 cosðÞ ~ q 0 0 0 0 0 0 00 00 0 0 0 e 26 3 27 6 7 6 76 7 6 7 6 76 7 f 0 sinðÞ ~ q 0 0 0 0 0 0 00 00 0 0 0 e 27 3 28 6 7 6 76 7 6 7 6 76 7 f 1 0 0 0 0 0 0 0 00 00 0 0 0 e 28 1 6 7 6 76 7 6 7 6 76 7 f e 8 1 0 0 0 0 0 0 0 00 00 0 0 0 25 6 7 6 76 7 6 7 6 76 7 f e 12 0 sinðÞ ~ q 0 0 0 0 0 0 00 00 0 0 0 8 6 7 6 3 76 7 4 5 4 56 7 f e 13 0 cosðÞ ~ q 0 0 0 0 0 0 00 00 0 0 0 12 3 6 7 4 5 f e 19 0 1 0 0 0 0 0 0 00 00 0 0 0 13 ~ ~ ~ ~ ~ ~ ~ ~ where h ¼ C h f and h ¼ C h f þ g ðÞ ~x f þ g ðÞ ~x with h ¼ h j ; 3 2 2 24 4 2 2 4 4 24 3 1 1 ðÞ ~xtðÞ;~utðÞ h ¼ h j ; g ðÞ ~x ¼ g ðÞ x j and g ðÞ ~x ¼ g ðÞ x j ; then the state variables 2 2 3 3 4 4 ðÞ ~xtðÞ;~utðÞ ðÞ ~xtðÞ;~utðÞ ðÞ ~xtðÞ;~utðÞ description can be obtained. 1 ∂e –cos (q ) ∂q ~ ~ p ,q 24 3 TF Sf:f (t) 10 MGY TF l:i ∂e i 27 –sin (q ) p ∂q 24 3 ~ p ,q 24 3 Figure 12. New causal paths with a source substituting a storage element. Mathematical and Computer Modelling of Dynamical Systems 53 sin (q ) TF q Se:e (t) 1 MTF TF Se:e (t) l:i ~ 1 –cos (q ) Figure 13. New causal paths with sources substituting the dissipation elements. ∂e 1 ~ sin(q ) –cos(q ) ~ ~ p ∂q 24 3 p ,q 24 3 43 TF Se:e (t) TF 26 38 3 47 q 1 MTF ~ 35 40 41 1 0 MGY 1 Sf:f (t) Se:e (t) TF 37 39 TF –cos(q ) 3 1 ∂e –sin(q ) p ~ ~ 42 49 ∂q p ,q 24 3 l:i R:R R:R l:i 1 1 2 24 l:i 1 4 R: i 22 25 MSe:S 1 20 1 2 ∂q MSe:S 2 1 1 2 ~ ~ δ 31 3 p ,q 24 3 34 1 ~ TF:cos(q ) C:C 7 9 MGY q 2 q δ δ MGY 3 3 C:C ∂φ (f ) l:i 17 1 x 27 30 GY:p ~ R:i ~ ~ 1 ∂p GY:p p ,q 24 12 33 24 3 ~ 10 27 11 TF 2 ~ 12 15 13 TF:h GY. 1 l:i 2 l:i 1 26 32 ∂φ (f ) ~ y 26 Sf:f TF:sin(q ) R:i 24 Sf:f 3 1 ~ ∂p ~ 2 24 p ,q 24 3 Figure 14. Linearized bond graph of the manipulator. This example illustrates the effectiveness of the proposed methodology to build a linearized bond graph and to determine the linearization of the class of non-linear systems considered in this paper. 6. Conclusions A procedure to linearize a class of non-linear physical systems in a bond graph approach is proposed. A linearized bond graph is obtained connecting the original bond graph that represents the non-linear system evaluated on the nominal trajectory with the bond graph representation of the new causal paths based on the linearization of the non-linear parts of the original bond graph. This paper permits consideration of storage elements in derivative and integral causality assignments and non-linear resistors. The effectiveness of the 54 G. Gonzalez Avalos and R. Galindo Orozco proposed methodology to obtain the linearized bond graphs and state variables lineariza- tion has been illustrated using an example. In [17], a bond graph model is linearized by replacing each non-linear element by a linearized element, hence this paper presents the advantage of obtaining the linearization in two steps: the original bond graph under the nominal trajectory and the additional bonds according to the causal paths that represent non-linear parts of the original bond graph model. Acknowledgement The authors would like to thank all the reviewers for their useful and valuable comments in improving this article. References [1] F.T. Brown, Engineering System Dynamics, Marcel Dekker, Inc, New York, 2001. [2] P.M. Anderson and A.A. Fouad, Power System Control and Stability, Science Press, Ames, IA, [3] P. Kundur, Power System Stability and Control, McGraw-Hill, New York, 1994. [4] P.J. Antsaklis and A.N. Michel, Linear Systems, Birkhauser, Boston, MA, 2006. [5] H. Wang and D. Linkens, Intelligent Supervisory Control, a Qualitative Bond Graph Reasoning Approach, World Scientific Publishing, Danvers, MA, 1996. [6] T. Kailath, Linear Systems, Prentice-Hall Inc, Englewood Cliffs, NJ, 1980. [7] C.-T. Chen, Linear System Theory and Design, Oxford University Press, New York, 1999. [8] J.S. Bay, Fundamentals of Linear State Space Systems, McGraw-Hill, Boston, MA, 1999. [9] P.C. Krause, O. Wasynczuk, and S.D. Sudhoff, Analysis of Electric Machinery, IEEE Press, Piscataway, NJ, 1995. [10] R.C. Rosenberg and D.C. Karnopp, Introduction to Physical System Dynamics, McGraw-Hill, New York, 1983. [11] W. Lin and H. Wang, Linearization techniques in fault diagnosis of non-linear systems, Proc. Instn Mech Engrs. 214 (part I) (2000), pp. 241–245. [12] D.C.H. Yang and S.W. Tzeng, Simplification and linearization of manipulator dynamics by the design of inertia distribution, Int. J. Rob. Res. 5 (1986), pp. 120–128. [13] Y. Shen, M.F. Golnaraghi, and G.R. Heppler, Analytical and experimental study of the response of a suspension system with a magnetorheological damper, J. Intell. Mater. Syst. Struct. 16 (2005), pp. 135–147. [14] D.C. Karnopp and R.C. Rosenberg, System Dynamics, Modeling and Simulation of Mechatronic Systems, Wiley, John & Sons, New York, 2000. [15] P.E. Wellstead, Physical System Modelling, Academic Press, London, 1979. [16] C. Sueur and G. Dauphin-Tanguy, Bond graph approach for structural analysis of MIMO linear systems, J. Flanklin Inst. 328 (1) (1991), pp. 55–70. [17] D. Karnopp, Power and energy in linearized physical systems, J. Franklin Inst. 303 (1) (1977), pp. 85–98. [18] P. Gawthrop and L. Smith, Metamodelling, Prentice-Hall, Herts, 1996. [19] W.J. Rugh, Linear System Theory, Prentice-Hall, Upper Saddle River, NJ, 1996. [20] P. Breedveld, A bond graph algorithm to determine the equilibrium state of a system,J. Franklin Inst. 318 (2) (1984), pp. 71–75. [21] E. Bideaux, F. Marquis, and S. Scavarda, Equilibrium set investigation using bicausality, Math. Comput. Model. Dyn. Syst. 12 (2006), spec. issue on Bond Graph Modelling, pp. 127–140. [22] B. Jean, Analysis and characterization of hybrid systems with Bond-Graphs, IEEE International Conference on SMC, pp. 264–269, Le Touquet, 17–20 October 1993. Mathematical and Computer Modelling of Dynamical Systems 55 Appendix. A proof of lemma FromðÞ 22 we have, ~ ~ fxðÞ ðÞt ;utðÞ¼ AxðÞxtðÞ þBxðÞutðÞ þHxðÞ ; u (46) and the linearized system defined byðÞ 23 and A matrix by ðÞ 7 , then the Jacobian is @f @ @ @ ~ ~ ~ ¼ AxðÞxtðÞ þ BxðÞutðÞ þ HxðÞ ; u (47) @x @x @x @x and consideringðÞ 14 ,1ðÞ6 andðÞ 21 , the first term of the right side ofðÞ 47 , the Jacobian is described by the submatrices of the junction structure, hi @ @ 1 1 E ðÞ x Ax; xtðÞ ¼ E ðÞ x S ðÞ x FxðÞ t @x @x 1 11 11 (48) þ E ðÞ x S ðÞ x MxðÞS ðÞ x FxðÞ t 12 21 @x hi 1 1 þ E ðÞ x S ðÞ x F ðÞ x FxðÞ t d 31 @x the term E ðÞ x S ðÞ x FxðÞ t can be written as follows: 2 32 3 11 1n α ðÞ x α ðÞ x S ðxÞ S ðÞ x 11 1n 11 11 6 . . . 76 7 . . . . . . . . . 4 54 5 . . . . . . n1 nn α ðÞ x α ðÞ x S ðxÞ S ðÞ x n1 nn 11 11 3 3 3 2 2 2 α ðxÞ α ðÞ x F F x 11 1n 1 11 1n 6 . . . 76 . 7 6 7 1 . . . . . . . . . . 4 54 5 E ðÞ x ¼ 4 5 ¼ . . . . . . . F F x α ðxÞ α ðÞ x n1 nn n n1 nn Where 2 3 2 3 α ðÞ x α ðÞ x 11 1n 6 . 7 6 . 7 ij ½ ðα ðÞ x Þ ðÞ α ðÞ x WithðÞ α ðÞ x ¼ . ;ðÞ α ðÞ x ¼ . and S ðÞ x is the element 4 5 4 5 1 n 1 n . . 11 α ðÞ x α ðÞ x n1 nn of S ðÞ x of row i and column j. The ExðÞ matrix has to be invertible and fromðÞ 15 it can be shown that ExðÞ is structurally invertible. However, ExðÞ can numerically be singular in a set of x. Hence, the previous term is: 82 3 S ðÞ x < 11 6 7 1 1 . E ðÞ x S ðÞ x FxðÞ t ¼ E ðÞ x 4 5ðÞ F x þþ F x 11 11 1 1n n n1 S ðÞ x 11 56 G. Gonzalez Avalos and R. Galindo Orozco 2 3 2 3 12 1n S ðÞ x S ðÞ x > 11 11 6 7 6 7 . . 6 7 6 7 .ðÞ F x þþ F xþþ .ðÞ F x þ þ F x ¼ 21 1 2n n n1 1 nn n 4 . 5 4 . 5 n2 nn S ðÞ x S ðÞ x 11 11 2 3 S ðÞ x h i 6 7 n n 1 . 6 7 E ðÞ x ðÞ S ðÞ x F x þ þðÞ S ðÞ x F x whereðÞ S ðÞ x ¼ . ; 11 1j j 11 nj j 11 1 j¼1 nj¼1 1 4 . 5 n1 S ðÞ x 2 3 1n S ðÞ x 6 7 6 7 ðÞ S ðÞ x ¼ . in a compact form; we have n 4 . 5 nn S ðÞ x n n 1 1 E ðÞ x S ðÞ x FxðÞ t ¼ E ðÞ x S ðÞ x Þ F x the Jacobian of this term is 11 11 ij j i¼1 ij¼1 @ @ n 1 1 n 1 E ðÞ x S ðÞ x FxðÞ t ¼ E ðÞ x ðÞ S ðÞ x F x þ E ðÞ x S ðÞ x F 11 11 ij j 11 i¼1 i j¼1 @x @x @ðÞ S ðÞ x n 11 1 i ¼ E ðÞ x F x þ ij j i¼1 @x j¼1 @ðÞ α n ik n 1 S ðÞ x F x þ E ðÞ x S ðÞ x F ij j 11 i¼1 11 j¼1 @x k¼1 @S ðÞ x @E ðÞ x ¼ E ðÞ x FxðÞ t þ S ðÞ x FxðÞ t @x @x þ E ðÞ x S ðÞ x F (49) hi hi @ðÞ S ðÞ x @S ðÞ x @ðÞ S ðÞ x @ðÞ S ðÞ x @E ðÞ x @ðÞ αðÞ x @ðÞ αðÞ x 11 11 11 11 i 1 n 1 n where ¼ and ¼ being and @x @x @x @x @x @x @x @ðÞ αðÞ x i 1 ; i.e., the Jacobians of column i of S ðÞ x and E ðÞ x , respectively. @x Similarly, the second term ofðÞ 48 is given by 11 11 @ @ S ðÞ x MxðÞS ðÞ x 1 11 11 1 12 21 E ðÞ x S ðÞ x MxðÞS ðÞ x FxðÞ t ¼ E ðÞ x FxðÞ t þ 12 21 @x @x @E ðÞ x 11 11 S ðÞ x MxðÞS ðÞ x FxðÞ t þ 12 21 @x 1 11 11 E ðÞ x S ðÞ x MxðÞS ðÞ x F (50) 12 21 and the last term ofðÞ 48 is @ @ S ðÞ x F S ðÞ x 14 31 1 1 1 E ðÞ x S ðÞ x F S ðÞ x FxðÞ t ¼ E ðÞ x FxðÞ t þ 14 31 @x @x Mathematical and Computer Modelling of Dynamical Systems 57 @E ðÞ x S ðÞ x F S ðÞ x FxðÞ t þ 14 31 @x 1 1 E ðÞ x S ðÞ x F S ðÞ x F (51) 14 31 by substitutingðÞ 49 ,5ðÞ0 andðÞ 51 intoðÞ 48 @ @AxðÞ ; x _ 1 1 1 E ðÞ x AxðÞ ; x _ xtðÞ ¼ E ðÞ x AxðÞ ; x _ þ E ðÞ x FxðÞ t (52) @x @x @E ðÞ x þ AxðÞ ; x _ FxðÞ t @x and in a similar way, the second term ofðÞ 47 is determined by @ @BxðÞ @E ðÞ x 1 1 E ðÞ x BxðÞutðÞ ¼ E ðÞ x utðÞ þ BxðÞutðÞ (53) @x @x @x the last term ofðÞ 47 is given by @ @HxðÞ ; u @E ðÞ x 1 1 E ðÞ x HxðÞ ; u ¼ E ðÞ x þ HxðÞ ; u (54) @x @x @x the Jacobians of the second and third terms ofðÞ 22 with respect to the input utðÞ are, @f @ S ðÞ x F S ðÞ x @HxðÞ ; u 14 31 1 1 B ¼ ¼ E ðÞ x BxðÞþ E ðÞ x z þ (55) @u @u @u evaluatingðÞ 52 ;ðÞ 53 ;ðÞ 54 andðÞ 55 under nominal trajectory fromðÞ 23 toðÞ 29 are proved.
Mathematical and Computer Modelling of Dynamical Systems – Taylor & Francis
Published: Jan 2, 2015
Keywords: bond graph; non-linear systems; linearization; DC motor
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