Power Scaling Bond Graph Approach to the. Passification of Mechatronic Systems - With Application to Electrohydraulic Valves

Transcription

1 Power Scaling Bond Graph Approach to the 1 Passification of Mechatronic Systems - With Application to Electrohydraulic Valves Perry Y. Li and Roger F. Ngwompo Abstract In many applications that require physical interaction with humans or other physical environments, passivity is a useful property to have in order to improve safety and ease of use. Many mechatronic applications (e.g. teleoperators, robots that interact with humans) fall into this category. In this paper, we develop an approach to design passifying control laws for mechatronic components from a bond graph perspective. Two new bond graph elements with power scaling properties are first introduced and the passivity property of bond graphs containing these elements are investigated. These elements are used to better model mechatronic systems that have embedded energy sources. A procedure for passifying mechatronic systems is then developed using the four way directional electrohydraulic flow control valve as an example. The passified valve is a two-port system that is passive with respect to the scaled power input at the command and hydraulic ports. This is achieved by representing the control valve in a suitable augmented bond graph, and then by replacing the signal bonds with power scaling elements. The procedure generalizes a previous passifying control law resulting in improved performance. Similar procedure can be applied to other mechatronic systems. Keyword: Passivity, bond graphs, power scaling, power scaling transformers / gyrators, electrohydraulics, man-machine systems. I. Introduction In the operation of systems requiring contacts with the environment or direct control by humans, passivity is an important property as it is related to both the safety and the ease to control the overall system. A passive system can be briefly described as a system that does not generate energy but only stores, dissipates, and releases it. The amount of energy that a passive system can impart to the environment is limited by the external input and so some safety is ensured compared to non-passive systems [1]. It also appears that because the concept of power can be used to plan and execute manipulation tasks, passive systems are potentially more user friendly. Research supported by the National Science Foundation under grant CMS P. Y. Li is with the Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, pli@me.umn.edu. R. F. Ngwompo is with the Department of Mechanical Engineering, University of Bath, Bath, BA2 7AY, United Kingdom, R.F.Ngwompo@bath.ac.uk

2 2 From the above observations, it would be helpful to use passive systems in tasks that require contacting the physical environment and/or direct control by humans. The passivity property of electromechanical systems have been exploited to develop overall control systems that are closed loop passive (see for example [2]). Although many hydraulic systems (e.g. in construction equipment) also involve direct human operation and direct contact, the passivity concept has only recently been applied to electro-hydraulic control systems [3]. In [3], the passivity property of the directional control valve was investigated from the controls perspective, and the valve was shown to be non-passive. Two alternative methods were proposed to make this device passive: by making structural or hardware redesign or by implementing active feedback compensation. Passified valves or other devices are useful since the design of additional passive control can be done much simply and more robustly for intrinsically passive systems than for arbitrary systems. For example, the actively feedback compensated passified valve in [3] enables the development of the first passive bilateral teleoperation scheme for electrohydraulic actuators [4]. Bond graph (see [5] for an introduction) is a physical approach to the modeling of physical systems that have increasingly been used in the analysis of systems for design and control ([6], [7], [8]). The inherent concept of power and energy embedded in bond graph representations suggests that this tool can be used to investigate the passivity property of systems and possibly provide alternate or generalized methods to make a system passive. The objective of this paper is to develop a general framework for analyzing passivity and developing passifying control laws for mechatronic systems using bond graph techniques. The electrohydraulic four-way directional control valve studied in [3] is used as an example to develop this procedure. From the full understanding of this example, a general procedure can be proposed to make non-passive systems passive. A key feature of many mechatronic systems is that they contain embedded power sources. The usefulness of these systems rely on the proper manipulation of the power delivered by these sources. Regular bond graphs, however, generally treat these power sources as external inputs. Also, any control or modulation are via the use of signal bonds that do not satisfy power continuity. Moreover, in multi-port systems, it would be useful for powers at different ports to take on different scalings. For example, a human operating a hydraulic excavator via a joystick exerts a much smaller power than the power that the excavator actually exerts. The power scaling concept is also not available in regular bond graphs. For these reasons, regular bond graphs are not adequate in addressing passivity and passification questions for these classes of mechatronic systems. In this paper, we propose two new bond graph elements, the power scaling transformers (PTF) and power scaling gyrators (PGY) to augment the bond graph framework. These are similar to regular transformers and gyrators but satisfy

3 3 a scaled power continuity. Power scaling bond graphs provide a framework to analyze the passivity of mechatronic systems with embedded energy sources and power scaling properties. This rest of the paper is organized as follows. In section II, the definition of passivity, its relationship to regular bond graphs, and a brief problem statement are given. In section III, two new bond graph elements are introduced and the passivity property of power scaling bond graphs investigated. Bond graph models of a directional control valve are presented in Section IV. The passifying algorithm for the valve, developed using the bond graph perspective is given in Section V. Some remarks regarding the generalization of the proposed bond graph is given in Section VI. Section VII contains some concluding remarks. II. Passivity and bond graph Given a dynamic system with input u and output y, a supply rate for the system can be defined to be any function s(u, y) R which, considered a function of time, is L 1 integrable for any finite time ( L 1e ). A system is said to be passive 1 [9] with respect to this supply rate s(u, y) if, for any given initial condition, there exists a constant c R so that for all time t and for all inputs u( ), t 0 s(u(τ), y(τ))dτ c 2. (1) Assume that the input u and output y are colocated effort and flow variables for a physical system, then a physically meaningful supply rate can be defined to be the inner product between the input and the output. This supply rate (with proper sign conventions) represents the power input into the system. In this case, the passivity condition (1) expresses the fact that for all input u( ) and the corresponding output y( ), no matter how the input is manipulated and how much time one waits, the maximum amount of energy that can be extracted from the system is limited by the constant c 2 (depends on initial conditions but not on time interval or inputs), which can be interpreted as the initial energy stored in the system. A standard regular bond graph [5] consists of interconnections of dissipative (R-), capacitive (C-) and inertance (I-) elements, transformers (TF), gyrators (GY), and their multi-port generalizations. Of these components, capacitive (C-) and inertance (I-) elements are energy conserving energy storage elements, whereas resistive (R-) elements are dissipative elements. Transformers and gyrators do not store or dissipate energy. Interconnections are made through power bonds ( ) or the 0- (common effort) or 1- (common flow) junctions via the colocated effort variables, ensuring power continuity. 1 Strictly speaking the term dissipative should be used instead unless s(u, y) is the pairing between a vector space and its dual. The supply rates we consider in this paper are indeed of this form.

4 4 Power generation is represented via effort sources (S e ) or (S f ) flow sources. These standard components and connections are suitable for physical systems. For example, Fig. 1 is a bond graph of a mass-damper system connected to a R-C circuit via a voice-coil transducer. The capacitive element is the electrical capacitor, and the inertance element is the inertia of the mass and the magnet in the voice-coil transducer. The direction of the half arrows ( ) denotes the direction of power flow given by the product of the effort (e) and flow (f) variables associated with the power bond. For the capacitive element C, we can define its displacement (or charge) q, the constitutive capacitance relation, and the energy storage W C by: q := C(e C ) := f C dt W C (e C ) := C(eC ) q=c(e 0 ) (2) e C (q)dq. where e C (q) := C 1 (q). Similarly, for the inertance element (I ), we can define its momentum p, the constitutive inertance relation, and the energy storage W I by: p := I(f I ) := e dt W I (f I ) := I(fI ) p=i(f 0 ) (3) f I (p)dp. where f I (p) := I 1 (p). C : R R and I : R R are possibly nonlinear one-to-one functions. For physical systems, it is always possible to choose appropriate datum e 0 and f 0 so that W C (e C ) and W I (f I ) are positive functions so that they represent physical energies. We assume that this is done in this paper. It can easily be shown that the C and I elements satisfy the power continuity relation (assuming power bonds point towards the elements): d dt W C = e C f C ; d dt W I = e I f I. For a resistive element, the constitutive relationship is (assuming power bonds point towards the elements): e R := R(f R ) ; e R f R 0 (4) where R : R R is a possibly nonlinear positive definite function. In more modern development, signal bonds in which either the flow or the effort variable is unilaterally transmitted (as signal) are also introduced in order to represent a wider class of mechatronic / control systems. These are represented as full arrows ( ) pointing in the direction where the signal is transmitted (see Fig. 10 for an example). Unlike power bonds, the source of the signal bond A in (A B) is unaffected by the destination of the bond, B. Unfortunately, because of this, signal

5 5 Se 2 Damper, R C I R Mass, I Se 1 Sf 1 R C R R2 0 i1 Sf 1 GY(m) 1 R1 i2 Se 1 0 i3 Se 2 Magnet / Voice coil Fig. 1. A regular bond graph with no active bonds or power scaling components. bonds do not satisfy power continuity. A signal bond can be considered a modulated effort or flow source acting on the bond graph portion downstream to the signal bond. Later in this paper, we shall introduce new elements that have power scaling properties between its ports. We first state an obvious result. Theorem 1 Consider a regular bond graph (such as the one in Fig. 1) with no active bonds or power scaling components. Suppose there are n i input (effort or flow) sources, n s C or I energy storage elements, and n R R dissipative elements. Let the sign convention be such that all input bonds point towards the bond graph, and power bonds point towards each C, I and R elements. Then, with respect to input bonds i 1, i 2,..., n i with effort and flow variables e ij, f ij, j = 1,..., k, the system is passive respect to the supply rate: s(f i1,..., f ik, e i1,..., e ik ) := k f ij e ij. (5) j=1 In order words, given a set of initial conditions, there exists c R s.t. for any inputs, and for any time t 0 t 0 s(f i1 (τ),..., f ini (τ), e i1 (τ),..., e ini (τ))dτ c 2. Let the total energies in all the capacitive and inertance elements be given by n s W total := j=1 W C/I,j

6 6 according to the constitutive equations (2), (3) for each C and I elements. Then, W total is a storage function for the bond graph and it satisfies: Ẇ total = D(f R1,..., f RnR, e R1,..., e RnR ) + s(f i1,..., f ini, e i1,..., e ini ) (6) where D(f R1,..., f RnR, e R1,..., e RnR ) := n R j=1 f Rj e Rj is the power dissipation in all the resistive elements. Proof: Let the storage function W total be the total energy as suggested in the theorem. Using the constitutive relationship of each C, I and R elements and the continuity of power in the junction structure and of the T F and GY elements, it is easy to show that: Ẇ total = n R n i e Rj f Rj + f ij e ij j=1 j=1 = D(f R1,..., f RnR, e R1,..., e RnR ) + s(f i1,..., f ini, e i1,..., e ini ) Ẇ total s(f i1,..., f ik, e i1,..., e ini ) (7) The last inequality is because D(f R1,..., f RnR, e R1,..., e RnR ) 0 which is a property of resistive elements. Integrating (7), and using the fact that W total (t) 0, we obtain the desired passivity property: W (0) = W (t) W (t = 0) t 0 s(f i1,..., f ini, e i1,..., e ini )dτ. Hence, a system that can be modeled by a regular bond graph (such as a physical system) is passive if all its ideal effort and flow sources (S e or S f ) are considered outside the system, and the supply rate is defined to be the total power input from these sources. In many control and mechatronic systems, however, the power source is unmodulated and is embedded in the system. The controller performs the conversion of this power from the sources. Therefore, a more meaningful way of looking at passivity would be in terms of the interactions of the system (with power sources embedded) with the controller (the algorithm), and with the external environment. The questions we are addressing in the subsequent sections are: a) how to appropriately represent this power modulation using bond graphs so that the passivity property of the control system can be investigated. b) how to determine a controller that makes the control system passive and how to represent the equivalent passive control system with bond graphs.

7 7 e1 f1 PTF(m, ρ) e2 f2 e1 f1 PTF(m, ρ ) e2 f2 e1 f1 PGY(r, ρ) e2 f2 e1 f1 PGY(r, ρ ) e2 f2 Fig. 2. Causal relations of power scaling transformers / gyrators. III. Power Scaling Transformers (PTF) and Power Scaling Gyrators (PGY) Before proceeding, we introduce two new bond graph elements: power scaling transformers and power scaling gyrators. The causal properties of these elements follow the regular two-port transformers and gyrators. The difference is that there is a possibly non-unity scaling factor that relates the power inputs at the two ports. Specifically, P T F (m, ρ) denotes a power scaling transformer with transformer modulus m and power scaling ρ in Fig. 2. Its effort and flow variables at the two ports are causally related by: e 1 := me 2 ; f 2 := (ρm)f 1 or e 2 := e 1 /m; f 1 := f 2 /(ρm). (8) As such, ρm is the kinematic scaling between the two flow variables. Similarly, for P GY (r, ρ), a power scaling gyrator with gyrator modulus r and power scaling ρ in Fig.2, the relationships between the effort and flow variables are: e 1 := rf 2 ; e 2 := (ρr)f 1 or f 1 := 1/(ρr)e 2 ; f 2 := (1/r)e 1. (9) For unity power scaling (i.e. ρ = 1) P T F (m, ρ) and P GY (r, ρ) reduce to regular transformers and gyrators. Notice for both P T F (m, ρ) and P GY (r, ρ), ρ(e 1 f 1 ) = (e 2 f 2 ), ρ. Thus, the power at input is scaled by the factor ρ before supplying it to the output. Proposition 1 A power scaling transformer P T F (m, ρ) or a power scaling gyrator P GY (r, ρ) are conserving with respect to the ρ-scaled power input in the sense that: s(f 1, f 2, e 1, e 2 ) := ρf 1 e 1 + ( f 2 e 2 ) = 0. (10) Here, the power directions are as shown in Fig. 2. Therefore, a power scaling transformer / gyrator is passive with s(,,, ) in (10) as the supply rate.

8 Proof: Using (8) and (9), the scaled passivity property of individual PTF and PGY can be obtained by directly verifying that s(f 1, f 2, e 1, e 2 ) in (10) does indeed vanish identically. 8 Definition 1 A bond graph with power scaling transformers / gyrators is said to be singly connected at a P T F (m, ρ) or P GY (r, ρ) if the graph is separated into two disjoint subgraphs when the power scaling element is removed. In other words, there should not be any loops containing the power scaling element. The following theorem states that a bond graph with power scaling elements has similar passivity property as a regular bond graph as long as it is singly connected at each non-unity power scaling element. Theorem 2 Moreover, if a bond graph with power scaling transformers / gyrators but no active bonds is singly connected at every non-unity power scaling transformer / gyrator, then with respect to the n i input bonds i 1, i 2,..., i ni (assuming all the sign convention of all input bonds correspond to power input into the system when the variables are positive), there exist power scalings ρ 1, ρ 2,..., ρ ni such that the system is passive respect to the supply rate: s(f i1,..., f ik, e i1,..., e ik ) := k ρ j f ij e ij. (11) j=1 In order words, given a set of initial conditions, there exists c s.t. for any inputs, and for any time t 0, Proof: t 0 s(f i1 (τ),..., f ini (τ), e i1 (τ),..., e ini (τ))dτ c 2. To show that a singly connected bond graph is passive with respect to a scaled supply rate, the proof procedure is illustrated in Fig. 3. First, remove all the k power scaling transformers and gyrators to form k m disjoint bond graphs. For bond graph i k m, associate an energy storage W i to be the sum of the storages of all the I and C elements in the bond graph. Let s i be the supply rate for each disjoint bond graph with respect to which it is passive. Now, recursively re-insert the k power scaling transformers and gyrators one-by-one, by combining two bond graphs at each step. This is so because the bond graph is singly connected. At each step, two passive systems, each represented by a bond graph, are connected, and a power scaling element is re-inserted. Consider step when the l k-th PTF/PGY is re-inserted. Suppose the storage functions of the two bond graphs to be re-connected to port 1 and port 2 of the transformer / gyrator are T l1, and T l2

9 9 C C1 R 0 R1 T11 = Wc1, s11=0 D11=fR1*eR1 I:1 PTF(r2, γ ) 2 I2 1 e1 PTF(r1, γ ) 1 Se T12 = WI2, s12=ei1*fi1 D12=0 R R2 0 Se e2 T22=0, s22=ei2*fi2 D22=fR2*eR2 γ T21 = T1 = 2 Wc1+WI2 S21=S1 = ei1*fi1 γ D21=D1 = 2 fr1*er W=T2 = γ ( γ Wc1+WI2) 1 2 S = S2 = γ 1γ (ei1*fi1)+ei2+fi2 D = D2 γ 1 2 (fr1*er)+(fr2*er2) Fig. 3. Example illustrating the proof procedure of Theorem 2. The sub-bond graphs are reconstituted and the storage functions, supply rates, and dissipation function as sub-graphs are combined are scaled and added up. respectively, their supply rates are s l1 and s l2, the dissipation rates are D l1 0 and D l2 0, and the power scaling of the transformer / gyrator is ρ l. Thus, for i = 1, 2, d dt T li = D li + s li s li. Notice that according to Lemma 1, these can be defined at the first step when all the power scaling components have been removed. Next, for the bond graph rejoined by the PTF or PGY, define the storage function of the combined bond graph to be T l = ρ l T l1 + T l2 and the supply rate to be s l := ρ l s l1 + s l2. Clearly, d dt T l = ρ l d dt T l1 + d dt T l2 = (ρd l1 + D l2 ) + (ρ s l1 + s l2 ) (ρ s l1 + s l2 ) =: s l (12) i.e. the combined bond graph is passive with s l as its the supply rate and T l as the storage function. Let l l + 1 and continue this process until the original bond graph is reconstituted. It is clear that the final supply rate is of the form (11) and the complete reconstituted bond graph is passive with respect to it. Remark: The condition for singly connectedness at the PTF/PGY serves to disallow loops that can cause positive feedback with sufficiently large loop gain. For example, the bond graph in Fig. 4 is not singly connected at the PTF. It dynamics are given by I d dt f 1 = (1 1/ρ)f 1 + u,

10 10 Se: u I f1 1 e1 f1 PTF(1, ρ) e2 f2 GY(1) Fig. 4. A non-passive bond graph with power scaling transformer that is not singly connected. (to tank) P 0 = 0 (from pump) P s B P P B A Q_B A Xv QA Connected to hydraulic actuator F P L = P P A B Q L = Q A = Q B Fig. 5. A typical four-way directional control valve. where u is the input effort. Therefore, the system is neither passive nor stable, when ρ > 1. The example in Fig. 4 also shows that the singly connectedness condition in Theorem 2 is not necessary, since for ρ < 1, the bond graph is passive with respect to the supply rate s := u f 1. In the rest of the paper, we illustrate, using an electro-hydraulic valve as an example, how bond graphs with power scaling components can be used to design passive mechatronic systems. The main idea is to develop passifying control laws so that the closed loop mechatronic system behaves like a singly connected bond graph with possibly power scaling components. IV. Bond graph models of a four-way directional control valve Figure 5 shows a typical critically centered, matched, four way directional control valve. By actuating the spool, the orifices in the valve are modulated to meter the out-going flow (Q A ) to the hydraulic actuator, and the return flow (Q B ) from it. Assuming the hydraulic actuator is flow conserving (e.g. in a double ended cylinder), and neglecting flow forces and valve chamber dynamics, then

11 11 x v x v R C C R Se:Ps 1 0 P A Q A Q B Load 0 1 (Actuator) P B Se:P0 Fig. 6. Bond graph of the hydraulic portion of the valve including fluid compressibility effects and interaction with load. This system is passive when the energy source is excluded from the system. Q L := Q A = Q B. A mathematical model of the valve is given by [10]: mẍ v = F (13) Q L (x v, P L ) = C dw ρ x v Ps sgn(x v )P L (14) where F is the total longitudinal force experienced by the spool, which can be controlled using an electromechanical / solenoid actuator; x v is the spool displacement; m is the spool inertia; C d and w are the discharge and area gradient coefficients of the valve; P s is the supply pressure; and P L is the load pressure (differential pressure between the actuator ports); sgn( ) denotes the sign function. Eq.(14) is derived by combining the orifice equations for the meter-in and meter-out orifices. It is applicable when sgn(x v )P L < P s, which is the usual scenario. A similar expression can be written for the common situation when sgn(x v )P L P s. The bond graph model for the valve can be decomposed into the spool dynamics part, and the hydraulics part. The spool dynamics is simply the dynamics of an inertia. A bond graph of the hydraulics portion, with the valve chamber dynamics included is shown in Figure 6. Notice that the valve displacement x v modulates the out-going flow from the pressure source to the load, and the return flow from the load to the reservoir, via the two orifices (modeled using R elements with parameters modulated by x v ) [5, p ]. This modulation connects the spool part and the hydraulic parts of the valve. A simplified model, with the assumptions of incompressible flow and that of the load being flow conserving (i.e. Q A = Q B = Q L ), corresponding to Eqs. (13)-(14), is shown in Figure 7 where P L = P A P B is the load pressure. From the perspective of control, a valve is a 2-port device that interacts with two external environments: the hydraulic load (via P L and Q L ) and the control system (via the valve command input), and the power supply is simply part of the system. From the model in Fig. 7, it is clear that whenever x v 0, it is possible to manipulate the load pressure P L so that the pump pressure source S e : P s

12 12 Passive? valve command Controller F.. m x v = F x v R Se:Ps 1 Q L Se:P L hydraulic port Fig. 7. Simplified bond graph of the valve. We wish to develop control law so that the system (with the energy source included) is passive as it interacts with the load and the command input. Q L Kq xv 1/K t P L Fig. 8. Equivalent electrical circuit for the hydraulic valve equation (15) which is equivalent to Eq. (14). delivers power to the external environment. In other words, as far as the hydraulic environment is concerned, the valve is not passive. Of course, the valve would be passive if S e : P s were also considered part of the external environment (Fig. 6). For this reason, despite its direct physical correspondence, the bond graph models in Figs. 6 and 7 are not convenient for the interpretation of passivity from the perspective of control. Following [3], an alternative representation that is more suitable for bond graph passivity analysis is obtained by first reformulating the flow equation (14) to be Q L (x v, P L ) = K q x v K t (x v, P L )P L (15) where K q = C d w P s /ρ > 0 and K t (x v, P L ) can be shown to be non-negative. Thus, we can think of the valve as being a flow source modulated by x v with a no-load flow gain K q in parallel with a nonlinear conductance K t (x v, P L ) that shunts flow (Fig. 8). The corresponding bond graph model is shown in Figure 9. Here, the spool inertia dynamics determines the spool displacement which in turn modulates a flow source with a gain K q. In contrast, in the bond graph in Fig. 7 using (14), the spool

13 13 I:m C:1 Passive? R: 1/Kt valve command Controller Se: F 1 d/dt(xv) 0 Sf 0 Se : P x L v :Kq Fig. 9. Active bond graph representation of 4-way directional control valve C:m I:1 R: 1/Kt Sf: F 0 1 Sf 0 d/dt(xv) xv :Kq Se : P L Fig. 10. Dualized active bond graph representation of 4-way directional control valve only modulates the resistance values, which only indirectly influence on the energy flow. In this new perspective, the goal of passification is to modulate the effort source S e : F with a feedback control so as to make the system appear passive to the external environment. V. Bond graph approach for passification Notice that the bond graph in Fig. 9 contains two signal bonds: one associated with the modulating effect of the spool displacement x v on the flow rate; the other associated with the integration of the spool velocity ẋ v to obtain the spool displacement x v. The main idea in our approach of passification of the valve is to replace these active signal bonds by passive power bonds or power scaling transformers / gyrators. We proceed in three steps: Step 0: Duality transformation Transforming the spool dynamics portion of the bond graph in Fig. 9 using the duality relationship, we obtain the bond graph in Fig. 10.

14 14 C:m I:1 R: 1/Kt R:1/B 0 z PTF(r2, γ 2 ) xv 1 PTF(r1, γ ) 1 0 Se : PL Se: Fx Fig. 11. Desired power scaling bond graph representation of 4-way directional control valve with bonds replaced by PTF / PGY. Step 1: Create a desired bond graph by first replacing active signal bonds and modulated effort / flow sources by power scaling transformers. The power scalings γ 1, γ 2 of the two PTF s and the modulation factor r 2 of the PTF that replaces the integrator signal bond are to be determined later. The modulation factor of r 1 of the PTF in the x v induced signal bond must be chosen to be K q /γ 1 to preserve the meaning of the flow variable at the 1 junction to remain to be x v. Notice that the dualization step in Step 0 can be avoided if we choose to replace the active bonds by P GY instead of P T F. We prefer to use P T F because they reduce to simple power bonds when both the modulation factor and the power scaling are unity. In this sense, they are more natural. Step 2: Add other regular or power scaling bond graph elements. One possibility is to add an effort source F x at the 1 node as an auxiliary control input, and add a R-element B at the left hand most 0 junction. The resulting bond graph is shown in Fig. 11. Notice that Fig. 11 is a bond graph with power scaling components but is singly connected at these components. Therefore by Theorem 2, the system represented by this bond graph is passive with respect to a supply rate: s(f x, P L, x v, Q L ) = γ 1 F xx v P L Q L. Step 3: Determine the appropriate spool dynamics that realize the desired bond graph. Label the effort variable in the 0 junction by z. Then, according to the bond graph in Fig. 11, the dynamics of x v and of z are given by: Notice that (16) provides the transformation z that is given by: ẋ v = 1 r 2 z + (F x r 1 P L ) (16) mż = 1 γ 2 r 2 x v Bz (17) z r 2 = ẋ v (F x r 1 P L ).

15 15 Differentiating (16) and utilizing (17), we obtain the spool dynamics necessary to realize the dynamics of the bond graph to be Substituting the expression for z, we have: mẍ v = m d dt (F x r 1 P L ) B z 1 x r 2 γ 2 r2 2 v. mẍ v = Bẋ v 1 x γ 2 r2 2 v B(F x r 1 P L ) + m d dt (F x r 1 P L ). (18) Comparison between (18) and (13) suggests that the ideal passifying control law should be of the form: F = Bẋ v Kx v + B(F x r 1 P L ) + m d dt (F x r 1 P L ), = Bẋ v Kx v r 1 BP L + BF x + m d dt (F x r 1 P L ) (19) where K = 1/ (γ 2 r 2 2). The first three terms of this control law are spool damping, centering spring, and pressure feedback which can be realized physically. In fact, fluid flow forces in the valve naturally induce centering spring force and damping [10]. So, control needs only augment to these. Pressure feedback can be obtained physically by modifying the spool geometry [3]. The fourth and last terms in (19) are the command forcing term, and the prediction term for dynamics cancellation. These must be provided by the control law. Consider now the closed loop system with (F x, x v ) and (P L, Q L ) as the input port variables. Following the proof of Theorem 2, we can choose as the storage function of the system, so that W = γ 1 γ 2 m 2 z2 + γ 1 2 x2 v (20) Ẇ = γ 1 γ 2 Bz 2 K t (x v, P L )P 2 L + (γ 1 F xx v P L Q L ) (21) Hence, the system is passive with respect to the supply rate: s valve (F x, P L, x v, Q L ) := γ 1 F xx v P L Q L. (22) where γ 1 F xx v represents power at the command port, and P L Q L represents the power at the hydraulic port. The second term in (21) represents the valve intrinsic energy dissipation due to the shunt conductance in K t (x v, P L ) in (15). The first term in (21) is however, dissipation which is an artifact of the proposed bond graph. It is therefore generally desirable that this term should be small.

16 16 C:m I:1 R: 1/Kt z R: 1/B γ 0 PTF(r2, 2 ) 1 PTF(r1, γ ) xv 1 0 Se : PL Sf: m Error Sf: Frob Se: Fx Fig. 12. Bond graph of passified valve with robustness modification and estimation error. Step 4: Adding robustness The control law requires estimating the derivative of F x r 1 P L. Generally, there will be an estimation error which can be considered flow source at the 0 junction. To combat its possible negative effect on passivity, we can add a dissipative term to ensure that the system dissipates more energy than it might possibly gain from the estimation error (Fig. 12). Assuming that we can estimate the bound for the estimation error: d dt (F x r 1 P L ) d dt (F x r 1 P L ) b err, (23) where d/dt( ) is the estimate of the derivative of the argument, the passifying control law can be modified to include the term: F rob = m sgn(z)b err. (24) This ensures that F rob z + Error z 0 for any estimation error Error( ) (the signal inside in (23)) satisfying its assumed bound. The penalty for using a conservative error bound in the robustness control term would be a large addition dissipation term γ 1 γ 2 mb err z in (21). Step 5: Choosing appropriate parameters The bond graph in Fig. 11 determines the nominal closed loop behavior of the passified valve. It is parameterized by (r 1, γ 1, r 2, γ 2, B). However, valve orifice relationship dictates that r 1 γ 1 = K q, the no load flow gain. Let γ > 0, A > 0 to be two constants and then define: as well as a transformed input F x := α F x so that r 1 := A α, γ 1 = K q r 1 r 2 := 1, γ 2 = 1 (25) γb, F x r 1 P L = 1 α [F x AP L ]. (26)

17 17 Then, the passifying control law (19), with the robustness term in (24) recovers exactly the active passifying control law in [3] by setting α = B. In this case, the passifying control (19) is parameterized by (r 1 = A/α, γ, B). α only plays the role of input scaling as in (26) but does not alter the passifying control for a given A/α. The possibility of arbitrary input scaling using α B is unknown in [3] without using the bond graph approach. Examination of the target bond graph dynamics (16)-(17) shows that r 2 does not play a role in term of x v dynamics is concerned. This, together with the constraint K q = r 1 γ 1 means that the passifying control law suggested by the bond graph in Fig. 11 is completely parameterized by (r 1 = A/α, γ, B). The closed loop transfer function of the valve passified using the parameters in (25) and the input scaling (26) is x v (s) = = s + B/m [s(s + B/m) + γb/m] [F x(s) r 1 P L (s)]. s + B/m α [s(s + B/m) + γb/m] [F x(s) AP L (s)]. (27) If we set both poles at s = B w, where B w signifies the bandwidth of the passified valve, we need B = 2mB w, and γ = B w /2. We get the spool dynamics and the output flow equation: x v (s) = s + 2B w α(s + B w ) 2 [F x(s) AP L (s)]. (28) Q L (x v, P L ) = K q x v K t (x v, P L )P L (15) If the bandwidth of operation is well below B w, the passified valve can be approximated by its static gain: x v 2 αb w [F x AP L ] z γx v = 1 2 B wx v This static approximation has facilitated the development of the first successful passive bilateral teleoperation of a hydraulic actuator [4]. When the static approximation holds, from (21), the dissipation in the valve is given by: Ẇ = 2α K q AB w z 2 K t (x v, P L )P 2 L + ([ ] ) Kq F x x v P L Q L A using z 1 2 B wx v during low frequency operation, Ẇ αb w 2 K q A x2 v K t (x v, P L )P 2 L + ([ ] ) Kq F x x v P L Q L A (29)

18 18 C:m I:1 R: 1/Kt Y(s) z 0 PTF(r2, γ 2 ) xv 1 PTF(r1, γ ) 1 0 Se : PL R: 1/B Sf: F1 TF( 1/α ) Z(s) 0 Se: Fx I: I f Y(s) Fig. 13. Alternate bond graph structures for passification Here, the second term represents the actual energy loss due to the shunt conductance in the valve (15), the last term is the supply rate consisting of the the control input power (K q /A)F x x v and the hydraulic output power P L Q L. The first term represents energy dissipation which is a consequence of our passification algorithm. In particular, for the same valve opening x v, the dissipation is proportional to αb w. In the teleoperation control in [4], this appears as extra damping in the haptic property of the control and adversely affects the way that human perceives and distinguishes the external environment. If α = B = 2B w m as in [3] is used, energy dissipation increases quadratically with bandwidth, thus presenting an apparent trade off between bandwidth and haptic property. With the extra flexibility afforded by α in the bond graph approach, the passification induced dissipation can be made arbitrarily small by simply adjusting α. VI. Generalization We already saw above that the use of α effectively removes the apparent tradeoff between bandwidth and dissipation. The bond graph approach also offers potential new ways to passify the valve. For example, target bond graph structure alternate to Fig. 11, such as Fig. 13 can be used. Here, a general admittance Y (s) as well as an additional input F 1 are attached to the 0 node, and a general impedance Z(s) and an input F x are attached to the 1 node. These flexibilities and the possibility of using dynamic elements can be used to shape the frequency response of the passified device and to improve the dissipation property. To illustrate this idea, consider the example in which a P-I control for the z variable is applied Y (s) = I f s + B; Z(s) = 0 : F 1 = 0

19 19 The bond graph for Y (s) is shown in Fig. 13. The valve dynamics become: ẋ v x v 1/α mż = K B 1 z + 0 (F x AP L ) (30) I f ω ω 0 where we have used the substitution z = z/r 2, K = 1/(γ 2 r 2 2), r 1 = A/α, and F x = αf x. This has the transfer function: x v (s) = 1 α(k I f + 1)s [ mif (K I f + 1) s(s + B/m) + (K I f + 1) mi f s(s + B/m) + (K I f + 1) ] (F x (s) AP L (s)) Both poles of the second order component inside the [ ] can be set at s = B w where B w is the desired bandwidth. The low frequency valve dynamics can be approximated by the integrator dynamics (take s 0) : ẋ v 1 α(k I f + 1) (F x AP L ). The integrator valve dynamics are attractive because command is now related to dq L /dt which is approximately proportional to actuator acceleration. The main advantage of this P-I passifying control is that the control does not dissipate any energy for low frequency operation, since z = r 2 z 0. This can be seen by using the storage function: ( If W mod := γ 1 γ 2 2 ω2 + m ) 2 z2 + γ 1 2 x2 v, (31) ([ ] ) Ẇ mod = γ 1 γ 2 Bz 2 K t (x v, P L )PL 2 Kq + F x x v P L Q L A ([ ] ) (32) K t (x v, P L )PL 2 Kq + F x x v P L Q L A Moreover, since the energy dissipation due to implementing the robustness term (24) is proportional to z, this will also vanish asymptotically. Therefore, the penalty for poorly estimating d dt (F x r 1 P L ) is also reduced. VII. Conclusion In this paper, a framework for deriving passifying control for mechatronic systems with embedded power sources, using power scaling bond graphs has been proposed. Power scaling bond graphs extend the regular bond graphs with the use of power scaling transformers and gyrators. These new elements capture both the concept of power scaling in bond graphs while maintaining the scaled power continuity which is essential for passivity analysis. It is shown that singly connected power scaling bond graphs are passive with respect to an appropriately scaled power input.

20 20 Passification control laws for mechatronic systems are obtained by defining the controls that would duplicate a target power scaling bond graph. Two key steps in defining the target bond graph are the modeling of the embedded power source using a modulated input, and the replacement of any signal bonds by power scaling transformers / gyrators. When the procedure was applied to an electrohydraulic valve as an example, it produces passifying control laws that generalize and improve over previous ones. Although the proposed procedure uses the electrohydraulic valve as an example, it should be applicable to other mechatronic systems as well. Current research is directed towards obtaining a tighter condition for when a power scaling bond graph is passive, and thus increasing the applicability of the proposed approach. References [1] J.E. Colgate, Coupled stability of multiport systems-theory and experiments., Transactions of the ASME. Journal of Dynamic Systems, Measurement and Control,, vol. 116, no. 3, pp , [2] Perry Y. Li and Roberto Horowitz, Control of Smart Exercise Machines: Part 1. Problem Formulation and Non-Adaptive Control, IEEE/ASME Transactions on Mechatronics, vol. 2, no. 4, pp , December [3] Perry Y. Li, Towards safe and human friendly hydraulics: the passive valve, ASME Journal of Dynamic Systems, Measurement and Control, vol. 122, no. 3, pp , [4] Perry Y. Li and Kailash Krishnaswamy, Passive bilateral teleoperation of a hydraulic actuator using an electrohydraulic passive valve, in Proceedings of 2001 American Control Conference, 2001, pp [5] Dean C. Karnopp, Donald L. Margolis, and R. C. Rosenberg, System Dynamics - Modeling and simulation of mechatronic systems, John Wiley and Sons Inc, [6] Roger F. Ngwompo, S. Scavada, and D. Thomasset, Inversion of linear time invaiant siso systems modelled by bond graph, Journal of the Franklin Institute, vol. 333(B), no. 2, pp , [7] Roger F. Ngwompo and Peter J. Gawthrop, Bond graph-based simulationof nonlinear inverse systems using physical performance specifications, Journal of the Franklin Institute, vol. 336, pp , [8] S. Y. Huang and K. Youcef-Toumi, Zero dynamics of physical systems from bond graph models -Part 1: SISO system, ASME Journal of Dynamical Systems Measurements and Control, vol. 121, pp , [9] J. C. Willems, Dissipative dynamical systems, part 1: General theory, Archive for Rational Mechanics and Analysis, pp , [10] Hebert E. Merritt, Hydraulic Control Systems, John Wiley and Sons, 1967.