With Application to Electrohydraulic Valves - Me.umn.edu
P. Y. Li Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455 e-mail: [email protected]
R. F. Ngwompo Department of Mechanical Engineering, University of Bath, Bath, BA2 7AY, United Kingdom e-mail: [email protected]
Power Scaling Bond Graph Approach to the Passification of Mechatronic Systems— With Application to Electrohydraulic Valves 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 properties 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. 关DOI: 10.1115/1.2101848兴 Keywords: passivity, bond graphs, power scaling, power scaling transformers/gyrators, electro-hydraulics, man-machine systems, mechatronics
1
Introduction
In the operation of systems requiring contacts with the environment or direct control by humans, passivity is an important property for 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 nonpassive systems 关1兴. Because the concept of “power” can be used to plan and execute manipulation tasks, passive systems are potentially more user friendly. For the above reasons, passive systems would be useful in tasks that require contacting the physical environment and/or direct control by humans. The passivity property of electromechanical systems has been exploited to develop overall control systems that are closed loop passive 共see, for example, 关2,3兴兲. Although many hydraulic systems 共e.g., in construction equipment兲 also involve direct human operation and direct contact, the passivity concept was not applied to electrohydraulic control systems until 关4兴. In 关4兴, the passivity property of the directional control valve was investigated from the controls perspective, and the valve was shown to be nonpassive. 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 more simply and more robustly for intrinsically passive systems than for arbitrary sysContributed by the Dynamic Systems, Measurement, and Control Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received June 28, 2003; final manuscript received March 28, 2005. Assoc. Editor: Prabhakar R. Pagilla.
tems. For example, the actively feedback compensated passified valve in 关4兴 enables the development of the first passive bilateral teleoperation scheme for electrohydraulic actuators 关5,6兴. Our objective of passifying a mechatronic system such as the valve in 关4兴 is somewhat different from the passification 共or passivation兲 of a system with a view towards stabilizing the system such as in 关7–10兴. In the latter, the desired passivity property is to find the control u = a共x , v兲, such that the system x˙ = f共x兲 + g共x兲u is passive with respect to the input-output pair 共v , y兲, where v is the redefined input injected through the control and y is an output function 共possible redefined via forward compensation兲. In the former, our objective is to manipulate some exogenous control input 共e.g., the valve input兲 so that the mechatronic system interacts with its existing, physical environment in a passive manner, i.e., the input-output pairs 共e.g., pressure and flow兲 are fixed and given. As such, we are interested in the passivity property which has a direct physical power interpretation. Bond graph 共see 关11兴 for an introduction兲 is a physical approach to the modeling of physical systems that have increasingly been used in the analysis of systems for design and control 关12–14兴. The inherent concept of power and energy embedded in bond graph representations, especially in regard to physical interaction with environment and between components, suggests that this tool can be used to investigate the passivity property of systems and possibly provide alternate or generalized methods to make mechatronic systems 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 关4兴 is used as an example to develop this proce-
Journal of Dynamic Systems, Measurement, and Control Copyright © 2005 by ASME
DECEMBER 2005, Vol. 127 / 633
Fig. 1 A regular bond graph with no active bonds or power scaling components
dure. From the full understanding of this example, a general procedure can be proposed to make nonpassive systems passive. The proposed approach is advantageous because it allows the physics of the mechatronic system to provide useful structural insights. This in turn makes the robustness property of the resulting passifying control more apparent. A key feature of many mechatronic systems is that they contain embedded power sources. The usefulness of these systems relies 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 is 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. This 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 共PTFs兲 and power scaling gyrators 共PGYs兲 to augment the bond graph framework. These are similar to regular transformers and gyrators but satisfy 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 Sec. 2, the definition of passivity, its relationship to regular bond graphs, and a brief problem statement are given. In Sec. 3, two new bond graph elements are introduced and the passivity property of power scaling bond graphs is investigated. Bond graph models of a directional control valve are presented in Sec. 4. The passifying algorithm for the valve, developed using the bond graph perspective, is given in Sec. 5. Some remarks regarding the generalization of the proposed bond graph are given in Sec. 6. Section 7 contains some concluding remarks.
2
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共u , y兲 苸 R which, considered a function of time, is L1 integrable for any 634 / Vol. 127, DECEMBER 2005
finite time 共苸L1e兲. A system is said to be passive1 关15兴 with respect to this supply rate s共u , y兲 if, for any given initial condition, there exists a constant c 苸 R so that for all time t and for all inputs u共·兲,
冕
t
s„u共t兲,y共兲…d 艌 − c2 .
共1兲
0
Assume that the input u and output y are collocated 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 共with proper sign conventions兲 represents the power input into the system. In this case, the passivity condition 共1兲 expresses the fact that for all input u共·兲 and the corresponding output y共·兲, 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 c2 共depends on initial conditions but not on time interval or inputs兲, which can be interpreted as the initial energy stored in the system. A standard regular bond graph 关11兴 consists of interconnections of dissipative 共R-兲, capacitive 共C-兲, and intertance 共I-兲 elements, transformers 共TFs兲, gyrators 共GYs兲, and their multi-port generalizations. Of these components, 共C-兲 and 共I-兲 elements are energy conserving energy storage elements; 共R-兲 are dissipative; and 共TFs兲 and 共GYs兲 do not store or dissipate energy. Interconnections are made through “power bonds” 共兲 or the “0-” 共common effort兲 or “1-” 共common flow兲 junctions via the collocated effort variables, ensuring power continuity. Power generation is represented via effort sources 共Se兲 or flow sources 共S f 兲. 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 共兲 denotes the direction of power flow given by the product of the effort 共e兲 and flow 共f兲 variables associated with the power bond. For the capacitive ele1 Strictly speaking the term “dissipative” should be used instead unless s共u , y兲 is the pairing between a vector space and its dual. The supply rates we consider in this paper are indeed of this form.
Transactions of the ASME
ment C-, we can define its “displacement” 共or charge兲 q, the constitutive capacitance relation, and the energy storage WC by q ª C共eC兲 ª
WC共eC兲 ª
冕
冕
ns
Wtotal ª
f C · dt, 共2兲 eC共q兲dq,
q=C共e0兲
where eC共q兲 ª C−1共q兲. Similarly, for the inertance element 共I-兲, we can define its “momentum” p, the constitutive inertance relation, and the energy storage WI by
WI共f I兲 ª
冕
冕
e · dt, 共3兲
I共 f I兲
f I共p兲dp,
p=I共 f 0兲
where f I共p兲 ª I−1共p兲. 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 e0 and f 0 so that WC共eC兲 and WI共f I兲 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 共assuming power bonds point to the elements兲: d W C = e C · f C, dt
eR · f R 艌 0,
共4兲
兺f e . j=1
ij ij
共5兲
In other words, given a set of initial conditions, there exists c 苸 R s.t. for any inputs, and for any time t 艌 0 t
0
s„f i1共兲, . . . , f in 共兲,ei1共兲, . . . ,ein 共兲…d 艌 − c2 . i
˙ W total = − D共f R1, . . . , f Rn ,eR1, . . . ,eRn 兲 + s共f i1, . . . , f in ,ei1, . . . ,ein 兲 R
R
i
i
共6兲 R f e where D共f R1 , . . . , f Rn , eR1 , . . . , eRn 兲 ª 兺nj=1 R j R j is the power disR R sipation in all the resistive elements. Proof. Let the storage function Wtotal be the total energy as suggested in the theorem. Using the constitutive relationship of each C-, I-, and R- element and the continuity of power in the junction structure and the TF and GY elements, it is easy to show that
nR
˙ W total = −
兺e j=1
ni
Rj f Rj
+
兺f e j=1
ij ij
= − D共f R1, . . . , f Rn ,eR1, . . . ,eRn 兲 R
R
+ s共f i1, . . . , f in ,ei1, . . . ,ein 兲 i
i
˙ ⇒W total 艋 s共f i1, . . . , f ik,ei1, . . . ,ein 兲,
共7兲
i
R
k
s共f i1, . . . , f ik,ei1, . . . ,eik兲 ª
according to the constitutive equations (2) and (3) for each C- and I- element. Then, Wtotal is a storage function for the bond graph and it satisfies
D共f R1, . . . , f Rn ,eR1, . . . ,eRn 兲 艌 0,
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 共as signal兲 are also introduced in order to represent a wider class of mechatronic/ control systems. These are represented as full arrows 共→兲 pointing in the direction where the signal is transmitted 共see Fig. 10 for an example兲. Unlike power bonds, the source of the signal bond A in 共A → B兲 is unaffected by the destination of the bond, B. Unfortunately, because of this, signal 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 the ports. We first state the following result for regular bond graphs which will be extended for bond graphs with power scaling elements. 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 ni input (effort or flow) sources, ns C- or I- energy storage elements, and nR R- dissipative elements. Let the sign convention be such that all input bonds point to the bond graph, and power bonds point to each of the C-, I-, and R- elements. Then, with respect to input bonds i1 , i2 , . . . , ni with effort and flow variables ei j , f i j , j = 1 , . . . , ni, the system is passive with respect to the supply rate:
冕
C/I,j
The last inequality is because
d WI = eI · f I . dt
For a resistive element, the constitutive relationship is 共assuming power bonds point to the elements兲 eR ª R共f R兲,
兺W j=1
C共eC兲
p ª I共f I兲 ª
Let the total energies in all the capacitive and inertance elements be given by
i
Journal of Dynamic Systems, Measurement, and Control
R
which is a property of resistive elements. Integrating 共7兲, and using the fact that Wtotal共t兲 艌 0, we obtain the desired passivity property: − W共0兲 = W共t兲 − W共t = 0兲 艋
冕
t
0
s共f i1, . . . , f in ,ei1, . . . ,ein 兲d . i
i
Hence, a system that can be modeled by a regular bond graph 共such as a physical system兲 is passive if all its ideal effort and flow sources 共Se or S f 兲 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 共with power sources embedded兲 with the controller 共the algorithm兲, and with the external environment. The questions being addressed in the subsequent sections are a. b.
how to appropriately represent this power modulation using bond graphs so that the passivity property of the control system can be investigated and how to determine a controller that makes the control system passive and how to represent the “equivalent” passive control system with bond graphs.
3 Power Scaling Transformers (PTFs) and Power Scaling Gyrators (PGYs) 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 nonunity scaling factor that relates the power inputs at the two ports. Specifically, PTF共m , 兲 denotes a power scaling transformer with transformer modulus m and power scaling in Fig. 2. Its effort and flow variables at the two ports are causally related by DECEMBER 2005, Vol. 127 / 635
Fig. 2 Causal relations transformers/gyrators
of
power
scaling
THEOREM 2. If a bond graph with power scaling transformers/ gyrators but no active bonds is singly connected at every nonunity power scaling transformer/gyrator, then with respect to the ni input bonds i1 , i2 , . . . , ini (assuming all the sign conventions of all input bonds correspond to power input into the system when the variables are positive), there exist power scalings 1 , 2 , . . . , ni such that the system is passive with respect to the supply rate: ni
e1 ª me2,
s共f i1, . . . , f in ,ei1, . . . ,ein 兲 ª
f 2 ª 共m兲f 1 , f 1 ª f 2/共m兲.
or e2 ª e1/m,
i
共8兲
As such, m is the kinematic scaling between the two flow variables. Similarly, for PGY共r , 兲, a power scaling gyrator with gyrator modulus r and power scaling in Fig. 2, the relationships between the effort and flow variables are e1 ª rf 2,
e2 ª 共r兲f 1 ,
or f 1 ª 1/共r兲e2,
f 2 ª 共1/r兲e1 .
共9兲
For unity power scaling 共i.e., = 1兲 PTF共m , 兲 and PGY共r , 兲 reduce to regular transformers and gyrators. Notice for both PTF共m , 兲 and PGY共r , 兲, 共e1 f 1兲 = 共e2 f 2兲. Thus, the power at input is scaled by the factor before supplying it to the output. PROPOSITION 1. A power scaling transformer PTF共m , 兲 or a power scaling gyrator PGY共r , 兲 are conserving with respect to the -scaled power input in the sense that s共f 1, f 2,e1,e2兲 ª f 1e1 + 共− f 2e2兲 = 0.
共10兲
Here, the power directions are as shown in Fig. 2. Therefore, a power scaling transformer/gyrator is passive with s共. , . , . , . 兲 in (10) as the supply rate. Proof. Using 共8兲 and 共9兲, the scaled passivity property of individual PTFs and PGYs can be obtained by directly verifying that 䊐 s共f 1 , f 2 , e1 , e2兲 in 共10兲 does indeed vanish identically. DEFINITION 1. A bond graph with power scaling transformers/ gyrators is said to be singly connected at a PTF共m , 兲 or PGY共r , 兲 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 a similar passivity property as a regular bond graph as long as it is singly connected at each nonunity power scaling element.
i
兺 f e . j=1
j ij ij
共11兲
In order words, given a set of initial conditions, there exist c s.t. for any inputs, and for any time t 艌 0,
冕
t
0
s„f i1共兲, . . . , f in 共兲,ei1共兲, . . . ,ein 共兲…d 艌 − c2 . i
i
Proof. 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 km disjoint bond graphs. For bond graph i 艋 km, associate an energy storage Wi to be the sum of the storages of all the I- and C- elements in the bond graph according to Theorem 1. Let si be the supply rate for each disjoint bond graph with respect to which it is passive. Now, recursively reinsert 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 reinserted. Consider the step when the l 艋 kth PTF/PGY is reinserted. Suppose the storage functions of the two bond graphs to be reconnected to ports 1 and 2 of the transformer/gyrator are Tl1, and Tl2, respectively, their supply rates are sl1 and sl2, the dissipation rates are Dl1 艌 0 and Dl2 艌 0, and the power scaling of the transformer/ gyrator is l. Thus, for i = 1 , 2, d Tli = − Dli + sli 艋 sli . dt Notice that according to Theorem 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 Tl = lTl1 + Tl2 and the supply rate to be sl ª lsl1 + sl2. Clearly,
Fig. 3 Example illustrating the proof procedure of Theorem 2. The subbond graphs are reconstituted and the storage functions, supply rates, and dissipation function as subgraphs are combined are scaled and added up.
636 / Vol. 127, DECEMBER 2005
Transactions of the ASME
Table 1 Dimensions in M„ass…, L„ength…, T„ime… of various variables
Fig. 4 A nonpassive bond graph with power scaling transformer that is not singly connected
d d d Tl = l Tl1 + Tl2 = − 共Dl1 + Dl2兲 + 共 · sl1 + sl2兲 dt dt dt 艋 共 · sl1 + sl2兲 ¬ sl ,
d f 1 = 共1 − 1/兲f 1 + u, dt
where u is the input effort. Therefore, the system is neither passive nor stable, when ⬎ 1. The example in Fig. 4 also shows that the singly connectedness condition in Theorem 2 is not necessary, since, for ⬍ 1, the bond graph is passive with respect to the supply rate s ª 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.
4 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 outgoing flow 共QA兲 to the hydraulic actuator and the return flow 共QB兲 from it. Assuming the hydraulic actuator is flow conserving 共e.g., in a double-ended cylinder兲, and neglecting flow forces and valve chamber dynamics, then QL ª QA = QB. A mathematical model of the valve is given by 关16兴 mx¨v = F,
Ps , PA , PB , PL
m
xv
F , Fx ,
Kq
关L3T−1兴
关ML−1T−2兴
关M兴
关L兴
关MLT−2兴
关L2T−1兴
1k , K
x*v = 共1kxv兲
r1
r2
␥1 , ␥2 , ␥
If
关MT−2兴
关MLT−2兴
关M −1L2T兴
关1兴
关1兴
关M −1T2兴
z , z⬘
Fx⬘
B
A
␣
Bw
关LT−1兴
关LT−1兴
关MT−1兴
关L2兴
关MT−1兴
关T−1兴
共12兲
i.e., the combined bond graph is passive with sl as its supply rate and Tl as its 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 共11兲 and the complete reconstituted bond graph is passive with respect to it. 䊐 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. Its dynamics are given by I
QL , QA , QB
共13兲
Fig. 5 A typical four-way directional control valve
Journal of Dynamic Systems, Measurement, and Control
QL共xv, PL兲 =
C dw x 冑Ps − sgn共xv兲PL , v
共14兲
where F is the total longitudinal force experienced by the spool, which can be controlled using an electromechanical/solenoid actuator; xv is the spool displacement; m is the spool inertia; Cd and w are the discharge and area gradient coefficients of the valve; Ps is the supply pressure; and PL is the load pressure 共differential pressure between the actuator ports兲; sgn共·兲 denotes the sign function. Equation 共14兲 is derived by combining the orifice equations for the meter-in and meter-out orifices. It is applicable when sgn共xv兲PL ⬍ Ps, which is the usual scenario. A similar expression can be written for the unusual situation when sgn共xv兲PL 艌 Ps. We now consider the bond graph representations of the valve. To facilitate understanding of the various bond graphs and transformations, the dimensions of the physical quantities and variables in standard M共ass兲, L共ength兲, and T共ime兲 units are tabulated in Table 1. The bond graph model for the valve can be decomposed into the spool dynamics portion and the hydraulics portion. 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 Fig. 6. Notice that the valve displacement xv modulates the outgoing flow from the pressure source to the load, and the return flow from the load to the reservoir, via the two orifices 共modeled using R- elements with parameters modulated by xv兲 关11, pp. 277–278兴. This modulation connects the spool portion and the hydraulic portion of the valve. A simplified model, with the assumptions of incompressible flow and that of the load being flow conserving 共i.e., QA = QB = QL兲, corresponding to Eqs. 共13兲 and 共14兲, is shown in Fig. 7 where PL = PA − PB is the load pressure. From the perspective of control, a valve is a two-port device that interacts with two external environments: the hydraulic load 共via PL and QL兲 and the control system 共via the valve command input兲, and the power supply is simply part of the system. From the model in Fig. 7, it is clear that whenever xv ⫽ 0, it is possible to manipulate the load pressure PL so that the pump pressure source Se : Ps delivers power to the external environment. In other
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.
DECEMBER 2005, Vol. 127 / 637
Fig. 10 Dualized active bond graph representation of four-way directional control valve
Fig. 7 Simplified bond graph of the valve. We wish to develop a control law so that the system „with the energy source included… is passive as it interacts with the load and the command input.
words, as far as the hydraulic environment is concerned, the valve is not passive. Of course, the valve would be passive if Se : Ps were also considered part of the external environment 共Fig. 6兲. 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 关4兴, an alternative representation that is more suitable for bond graph passivity analysis is obtained by first reformulating the flow equation 共14兲 to be
In this new perspective, the goal of passification is to modulate the effort source Se : F with a feedback control so as to make the system appear passive to the external environment.
5
Bond Graph Approach for Passification
Fig. 8 Equivalent electrical circuit for the hydraulic valve equation „15… which is equivalent to Eq. „14…
We now proceed to develop a control law to passify the electrohydraulic valve. This involves several manipulations and transformations of the bond graph in Fig. 9. Please refer to Table 1 for the dimensions of the various variables. Notice that the bond graph in Fig. 9 contains two signal bonds: one associated with the modulating effect of the spool displacement xv on the flow rate, the other associated with the integration of the spool velocity x˙v to obtain the spool displacement xv. 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, of which step 0 is optional. 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. In this step, the “0” and “1” junctions, C- and I- elements, effort and flow variables, and the direction of the causality are exchanged to obtain an equivalent bond graph. This step is optional and it is used so that the resulting power scaling bond graphs to be developed later will have transformer-type instead of gyrator-type causalities. 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 ␥1, ␥2 of the two PTF’s and the modulation factor r2 of the PTF that replaces the “integrator” signal bond are to be determined later. The modulation factor of r1 of the PTF in the xv induced signal bond must be chosen so that Kq = 1kr1␥1 to preserve the relationship between the no-load flow and xv in 共15兲. Notice that if the dualization step in step 0 is omitted, then the active bond on the right-hand side should be replaced by PGY instead of PTF. We prefer to use PTF because they reduce to simple power bonds when both the modulation factor and the power scaling are unity, as opposed to reducing to GY in the case of PGY when the duality transformation is omitted. Whether one dualizes first is simply a stylistic issue. 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 lefthand most “0” junction. The resulting bond graph is shown in Fig. 11. Notice that Fig. 11 is a bond graph with power scaling com-
Fig. 9 Active bond graph representation of four-way directional control valve. 1k is the unit fictitious stiffness „dimension †MT−2‡… associated with the integral relationship between x˙v and xv. xv* ª „1kxv… is the effort variable at the 0 junction.
Fig. 11 Desired power scaling bond graph representation of four-way directional control valve with bonds replaced by PTF/PGY
QL共xv, PL兲 = Kqxv − Kt共xv, PL兲PL ,
共15兲
where Kq = Cdw冑Ps / ⬎ 0 and Kt共xv , PL兲 can be shown to be nonnegative. Thus, we can think of the valve as being a flow source modulated by the spool displacement xv with a no-load flow gain Kq with respect to xv, in parallel with a nonlinear conductance Kt共xv , PL兲 that shunts flow 共Fig. 8兲. The corresponding bond graph model is shown in Fig. 9. Here, the time integration of x˙v to form xv is modeled using a mechanical capacitive element 共e.g., a spring兲 represented by a compliance of 1 / 1k where 1k is the unit stiffness 共i.e., has a value of 1兲. The effort variable at the 0 junction is x*v ª 共1kxv兲 共has the dimension of a force 关MLT−2兴兲. It has the same value as xv关M兴 but with a change in units 共Table 1兲. In Fig. 9, the spool inertia dynamics determines the spool displacement which in turn modulates a flow source with a gain Kq. In contrast, in the bond graph in Fig. 7 using 共14兲, the spool only modulates the resistance values, which has only indirect influence on the energy flow.
638 / Vol. 127, DECEMBER 2005
Transactions of the ASME
ponents 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共F⬘x , PL,xv,QL兲 = 共1k␥1兲F⬘x xv − PLQL . 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 xv and of z are given by x˙v =
1 z + 共F⬘x − r1 PL兲, r2
mz˙ = −
1k x − Bz. ␥ 2r 2 v
共16兲 共17兲
Notice that 共16兲 provides the transformation z that is given by z = x˙v − 共F⬘x − r1 PL兲. r2 Differentiating 共16兲 and utilizing 共17兲, we obtain the spool dynamics necessary to realize the dynamics of the bond graph to be mx¨v = m
Substituting the expression for z, we have d − r1 PL兲. mx¨v = − Bx˙v − x − r1 PL兲 + m 共F⬘ 2 xv − B共F⬘ dt x ␥ 2r 2 共18兲 Comparison between 共18兲 and 共13兲 suggests that the ideal passifying control law should be of the form
= − Bx˙v − Kxv − r1BPL + BF⬘x + m
d 共F⬘ − r1 PL兲, dt x
共19兲
where K = 1k / 共␥2r22兲. 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 关16兴. So, control needs only augment to these. Pressure feedback can be obtained physically by modifying the spool geometry 关4兴. The fourth and last terms in 共19兲 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 共F⬘x , xv兲 and 共PL , QL兲 as the input port variables. Following the proof of Theorem 2, we can choose 1 k␥ 1 2 m W = ␥ 1␥ 2 z 2 + x 2 v 2
共20兲
as the storage function of the system, so that ˙ = − ␥ ␥ Bz2 − K 共x , P 兲P2 + 共共1 ␥ 兲F⬘x − P Q 兲. 共21兲 W 1 2 t v L L k 1 x v L L Hence, the system is passive with respect to the supply rate: svalve共F⬘x , PL,xv,QL兲 ª 共1k␥1兲F⬘x xv − PLQL ,
冏
冏
ˆd d 共F⬘x − r1 PL兲 − 共F⬘x − r1 PL兲 艋 berr , dt dt
Frob = − m · sgn共z兲berr .
1k
d 共F⬘ − r1 PL兲 dt x
The control law requires estimating the derivative of F⬘x − r1 PL. 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 共Fig. 12兲. Assuming that we can estimate the bound for the estimation error,
共23兲
where dˆ / dt共·兲 is the estimate of the derivative of the argument, the passifying control law can be modified to include the term
1k d z 共F⬘x − r1 PL兲 − B − xv . dt r2 ␥2r22
F = − Bx˙v − Kxv + B共F⬘x − r1 PL兲 + m
Fig. 12 Bond graph of passified valve with robustness modification and estimation error
共22兲
where 共1k␥1兲F⬘x xv represents the power at the command port, and PLQL represents the power at the hydraulic port. The second term in 共21兲 represents the valve intrinsic energy dissipation due to the shunt conductance in Kt共xv , PL兲 in 共15兲. The first term in 共21兲 is the dissipation, which is in fact an artifact of the proposed bond graph. This term is generally desired to be small. Step 4. Adding robustness. Journal of Dynamic Systems, Measurement, and Control
共24兲
This ensures that Frobz + Error· z 艋 0 for any estimation error Error共·兲 关the signal inside 兩·兩 in 共23兲兴 satisfying its assumed bound. The penalty for using a conservative error bound in the robustness control term would be a large additional dissipation term −␥1␥2mberr兩z兩 in 共21兲. 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 共r1 , ␥1 , r2 , ␥2 , B兲. However, valve orifice relationship dictates that 共1k␥1兲r1 = Kq, the no load flow gain. Let ␥ ⬎ 0, A ⬎ 0 be two constants and then define r1 ª
A , ␣
r2 ª 1,
␥1 =
Kq , r1
1 ␥2 = , ␥B
共25兲
as well as a transformed input Fx ª ␣ · F⬘x so that F⬘x − r1 PL =
1 共Fx − APL兲. ␣
共26兲
Then, the passifying control law 共19兲, with the robustness term in 共24兲, recovers exactly the active passifying control law in 关4兴 by setting ␣ = B. In this case, the passifying control 共19兲 is parameterized by 共r1 = A / ␣ , ␥ , B兲. ␣ only plays the role of input scaling as in 共26兲 but does not alter the passifying control for a given A / ␣. The possibility of arbitrary input scaling using ␣ ⫽ B is unknown in 关4兴 without using the bond graph approach. Examination of the target bond graph dynamics 共16兲 and 共17兲 shows that r2 does not play a role in terms of xv dynamics. This, together with the constraint Kq = 1k␥1r1, means that the passifying control law suggested by the bond graph in Fig. 11 is completely parameterized by 共r1 = A / ␣ , ␥ , B兲. The closed loop transfer function of the valve passified using the parameters in 共25兲 and the input scaling 共26兲 is DECEMBER 2005, Vol. 127 / 639
xv共s兲 = =
s + B/m 关F⬘共s兲 − r1 PL共s兲兴 关s共s + B/m兲 + ␥B/m兴 x s + B/m 关Fx共s兲 − APL共s兲兴. ␣关s共s + B/m兲 + ␥B/m兴
共27兲
If we set both poles at s = −Bw, where Bw signifies the bandwidth of the passified valve, we need B = 2mBw and ␥ = Bw / 2. We get the spool dynamics and the output flow equation: xv共s兲 =
s + 2Bw 关Fx共s兲 − APL共s兲兴, ␣共s + Bw兲2
共28兲
QL共xv, PL兲 = Kqxv − Kt共xv, PL兲PL .
共15⬘兲
Fig. 13 Alternate bond graph structures for passification
If the bandwidth of operation is well below Bw, the passified valve can be approximated by its static gain: xv ⬇
Y共s兲 =
2 共Fx − APL兲, ␣Bw
冢 冣冢
1
x˙v
This static approximation has facilitated the development of the first successful passive bilateral teleoperation of a hydraulic actuator 关5,6兴. When the static approximation holds, from 共21兲, the dissipation in the valve is given by
冋冉 冊
册
Kq F xx v − P LQ L ; A
mz˙⬘
˙ ⬇ − ␣Bw Kq x2 − K 共x , P 兲P2 + W t v L L 2 A v
冋冉 冊
˙ If
xv共s兲 =
0
1
0
= −K −B −1 0 1 0
册
Here, the second term represents the actual energy loss due to the shunt conductance in the valve 共15兲; the last term is the supply rate consisting of the control input power 共Kq / A兲Fxxv and the hydraulic output power PLQL. The first term represents energy dissipation which is a consequence of our passification algorithm. In particular, for the same valve opening xv, the dissipation is proportional to ␣Bw. In the teleoperation control in 关5,6兴, this appears as an extra damping in the haptic property of the controlled system and adversely affects the way that a human perceives and distinguishes the external environment. If ␣ = B = 2Bwm as in 关4兴 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 ␣ in the bond graph approach, the passificationinduced dissipation can be made arbitrarily small by simply adjusting ␣.
z⬘
+
0
共Fx − APL兲, 共30兲
0
冋
mI f 共KI f + 1兲s共s + B/m兲 + 共KI f + 1兲 1 ␣共KI f + 1兲s mI f s共s + B/m兲 + 共KI f + 1兲
册
Generalization
We already saw above that the use of ␣ effectively removes the apparent trade-off between bandwidth and dissipation. The bond graph approach also offers potential new ways to passify the valve. For example, target bond graph structure alternates to Fig. 11, such as Fig. 13, can be used. Here, a general admittance Y共s兲 as well as an additional input F1 are attached to the “0” node, and a general impedance Z共s兲 and an input Fx 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:
共31兲
Both poles of the second-order component inside the 关·兴 can be set at s = −Bw where Bw is the desired bandwidth. The low-frequency valve dynamics can be approximated by the integrator dynamics 共take s → 0兲: x˙v ⬇
1 共Fx − APL兲. ␣共KI f + 1兲
The integrator valve dynamics are attractive because command is now related to dQL / 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 = r2z⬘ ⬇ 0. This can be seen by using the storage function: Wmod ª ␥1␥2
冉
冊 冋冉 冊
1 k␥ 1 2 If 2 m 2 x , + z + 2 v 2 2
2 2 ˙ W mod = − ␥1␥2Bz − Kt共xv, PL兲PL +
⬇ − Kt共xv, PL兲PL2 +
640 / Vol. 127, DECEMBER 2005
冣冢 冣 冢 冣 1/␣
xv
⫻关Fx共s兲 − APL共s兲兴.
Kq F xx v − P LQ L . A 共29兲
6
F1 = 0.
where we have used the substitution z⬘ = z / r2, K = 1k / 共␥2r22兲, r1 = A / ␣, and Fx = ␣F⬘x . This has the transfer function:
1
using z ⬇ − 2 Bwxv during low frequency operation,
Z共s兲 = 0,
The bond graph for Y共s兲 is shown in Fig. 13. The valve dynamics become
z ⬇ − ␥ x v = − 2 B wx v .
˙ = 2␣ Kq z2 − K 共x , P 兲P2 + W t v L L ABw
If + B, s
冋冉 冊
Kq F xx v − P LQ L A
册
Kq F xx v − P LQ L . A
共32兲
册 共33兲
Moreover, since the energy dissipation due to implementing the robustness term 共24兲 is proportional to 兩z兩, this will also vanish asymptotically. Therefore, the penalty for poorly estimating 共d / dt兲共F⬘x − r1 PL兲 is also reduced.
7
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 Transactions of the ASME
analysis. It is shown that singly connected power scaling bond graphs are passive with respect to an appropriately scaled power input. 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 electro-hydraulic 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 to obtaining a tighter condition to determine when a power scaling bond graph is passive, and thus increasing the applicability of the proposed approach.
Acknowledgment Research supported by the National Science Foundation under Grant No. CMS-0085230.
References 关1兴 Colgate, J. E., 1994, “Coupled Stability of Multiport Systems-Theory and Experiments,” ASME J. Dyn. Syst., Meas., Control, 116共3兲, pp. 419–428. 关2兴 Li, P. Y., and Horowitz, R., 1997, “Control of Smart Exercise Machines: Part 1. Problem Formulation and Non-Adaptive Control,” IEEE/ASME Trans. Mechatron., 2共4兲, pp. 237–247. 关3兴 Lee, D. J., and Li, P. Y., 2003, “Passive Feedforward Approach for Linear Dynamically Similar Bilateral Teleoperated Manipulators,” IEEE Trans. Rob.
Journal of Dynamic Systems, Measurement, and Control
Autom., 19共13兲, pp. 443–456. 关4兴 Li, P. Y., 2000, “Towards Safe and Human Friendly Hydraulics: The Passive Valve,” ASME J. Dyn. Syst., Meas., Control, 122共3兲, pp. 402–409. 关5兴 Li, P. Y., and Krishnaswamy, K., 2001, “Passive Bilateral Teleoperation of a Hydraulic Actuator Using an Electrohydraulic Passive Valve,” in Proceedings of 2001 American Control Conference, pp. 3932–3937. 关6兴 Li, P. Y., and Krishnaswamy, K., 2004, “Passive Bilateral Teleoperation of a Hydraulic Actuator Using an Electrohydraulic Passive Valve,” Int. J. Fluid Power, 13, pp. 43–56. 关7兴 Seron, M. M., Hill, D. J., and Fradkov, A. L., 1994, “Adaptive Passification of Nonlinear Systems,” in Proceedings of the 33rd Conference on Decision and Control, pp. 190–195. 关8兴 Byrnes, C. I., Isidori, A., and Willems, J. C., 1991, “Passivity, Feedback Equivalence, and the Global Stabilization of Minimum Phase Nonlinear Systems,” IEEE Trans. Autom. Control, 36共11兲, 1228–1240. 关9兴 Kelkar, A. G., and Joshi, S. M., 1997, “Robust Control of Non-Passive Systems via Passification,” in Proceedings of 1997 American Control Conference, pp. 2657–2661. 关10兴 Song, Y. I., Shim, H., and Seo, J. H., 1999, “Passification of Nonlinear Systems via Dynamic Feedback,” in Proceedings of 36th Conference on Decision and Control, Vol. 5, pp. 4877–4878. 关11兴 Karnopp, D. C., Margolis, D. L., and Rosenberg, R. C., 2000, System Dynamics—Modeling and Simulation of Mechatronic Systems, John Wiley and Sons Inc. New York. 关12兴 Ngwompo, R. F., Scavada, S., and Thomasset, D., 1996, “Inversion of Linear Time Invariant Siso Systems Modelled by Bond Graph,” J. Franklin Inst., 333(B)共2兲, pp. 157–174. 关13兴 Ngwompo, R. F., and Gawthrop, P. J., 1999, “Bond Graph-Based Simulation of Nonlinear Inverse Systems Using Physical Performance Specifications,” J. Franklin Inst., 336, pp. 1225–1247. 关14兴 Huang, S. Y., and Youcef-Toumi, K., 1999, “Zero Dynamics of Physical Systems From Bond Graph Models-Part 1: SISO System,” ASME J. Dyn. Syst., Meas., Control, 121共1兲, pp. 19–26. 关15兴 Willems, J. C., 1972, “Dissipative Dynamical Systems, Part 1: General Theory,” Arch. Ration. Mech. Anal., 45, pp. 321–351. 关16兴 Merritt, H. E., 1967, Hydraulic Control Systems, John Wiley and Sons, New York.
DECEMBER 2005, Vol. 127 / 641
R. F. Ngwompo Department of Mechanical Engineering, University of Bath, Bath, BA2 7AY, United Kingdom e-mail: [email protected]
Power Scaling Bond Graph Approach to the Passification of Mechatronic Systems— With Application to Electrohydraulic Valves 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 properties 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. 关DOI: 10.1115/1.2101848兴 Keywords: passivity, bond graphs, power scaling, power scaling transformers/gyrators, electro-hydraulics, man-machine systems, mechatronics
1
Introduction
In the operation of systems requiring contacts with the environment or direct control by humans, passivity is an important property for 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 nonpassive systems 关1兴. Because the concept of “power” can be used to plan and execute manipulation tasks, passive systems are potentially more user friendly. For the above reasons, passive systems would be useful in tasks that require contacting the physical environment and/or direct control by humans. The passivity property of electromechanical systems has been exploited to develop overall control systems that are closed loop passive 共see, for example, 关2,3兴兲. Although many hydraulic systems 共e.g., in construction equipment兲 also involve direct human operation and direct contact, the passivity concept was not applied to electrohydraulic control systems until 关4兴. In 关4兴, the passivity property of the directional control valve was investigated from the controls perspective, and the valve was shown to be nonpassive. 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 more simply and more robustly for intrinsically passive systems than for arbitrary sysContributed by the Dynamic Systems, Measurement, and Control Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received June 28, 2003; final manuscript received March 28, 2005. Assoc. Editor: Prabhakar R. Pagilla.
tems. For example, the actively feedback compensated passified valve in 关4兴 enables the development of the first passive bilateral teleoperation scheme for electrohydraulic actuators 关5,6兴. Our objective of passifying a mechatronic system such as the valve in 关4兴 is somewhat different from the passification 共or passivation兲 of a system with a view towards stabilizing the system such as in 关7–10兴. In the latter, the desired passivity property is to find the control u = a共x , v兲, such that the system x˙ = f共x兲 + g共x兲u is passive with respect to the input-output pair 共v , y兲, where v is the redefined input injected through the control and y is an output function 共possible redefined via forward compensation兲. In the former, our objective is to manipulate some exogenous control input 共e.g., the valve input兲 so that the mechatronic system interacts with its existing, physical environment in a passive manner, i.e., the input-output pairs 共e.g., pressure and flow兲 are fixed and given. As such, we are interested in the passivity property which has a direct physical power interpretation. Bond graph 共see 关11兴 for an introduction兲 is a physical approach to the modeling of physical systems that have increasingly been used in the analysis of systems for design and control 关12–14兴. The inherent concept of power and energy embedded in bond graph representations, especially in regard to physical interaction with environment and between components, suggests that this tool can be used to investigate the passivity property of systems and possibly provide alternate or generalized methods to make mechatronic systems 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 关4兴 is used as an example to develop this proce-
Journal of Dynamic Systems, Measurement, and Control Copyright © 2005 by ASME
DECEMBER 2005, Vol. 127 / 633
Fig. 1 A regular bond graph with no active bonds or power scaling components
dure. From the full understanding of this example, a general procedure can be proposed to make nonpassive systems passive. The proposed approach is advantageous because it allows the physics of the mechatronic system to provide useful structural insights. This in turn makes the robustness property of the resulting passifying control more apparent. A key feature of many mechatronic systems is that they contain embedded power sources. The usefulness of these systems relies 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 is 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. This 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 共PTFs兲 and power scaling gyrators 共PGYs兲 to augment the bond graph framework. These are similar to regular transformers and gyrators but satisfy 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 Sec. 2, the definition of passivity, its relationship to regular bond graphs, and a brief problem statement are given. In Sec. 3, two new bond graph elements are introduced and the passivity property of power scaling bond graphs is investigated. Bond graph models of a directional control valve are presented in Sec. 4. The passifying algorithm for the valve, developed using the bond graph perspective, is given in Sec. 5. Some remarks regarding the generalization of the proposed bond graph are given in Sec. 6. Section 7 contains some concluding remarks.
2
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共u , y兲 苸 R which, considered a function of time, is L1 integrable for any 634 / Vol. 127, DECEMBER 2005
finite time 共苸L1e兲. A system is said to be passive1 关15兴 with respect to this supply rate s共u , y兲 if, for any given initial condition, there exists a constant c 苸 R so that for all time t and for all inputs u共·兲,
冕
t
s„u共t兲,y共兲…d 艌 − c2 .
共1兲
0
Assume that the input u and output y are collocated 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 共with proper sign conventions兲 represents the power input into the system. In this case, the passivity condition 共1兲 expresses the fact that for all input u共·兲 and the corresponding output y共·兲, 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 c2 共depends on initial conditions but not on time interval or inputs兲, which can be interpreted as the initial energy stored in the system. A standard regular bond graph 关11兴 consists of interconnections of dissipative 共R-兲, capacitive 共C-兲, and intertance 共I-兲 elements, transformers 共TFs兲, gyrators 共GYs兲, and their multi-port generalizations. Of these components, 共C-兲 and 共I-兲 elements are energy conserving energy storage elements; 共R-兲 are dissipative; and 共TFs兲 and 共GYs兲 do not store or dissipate energy. Interconnections are made through “power bonds” 共兲 or the “0-” 共common effort兲 or “1-” 共common flow兲 junctions via the collocated effort variables, ensuring power continuity. Power generation is represented via effort sources 共Se兲 or flow sources 共S f 兲. 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 共兲 denotes the direction of power flow given by the product of the effort 共e兲 and flow 共f兲 variables associated with the power bond. For the capacitive ele1 Strictly speaking the term “dissipative” should be used instead unless s共u , y兲 is the pairing between a vector space and its dual. The supply rates we consider in this paper are indeed of this form.
Transactions of the ASME
ment C-, we can define its “displacement” 共or charge兲 q, the constitutive capacitance relation, and the energy storage WC by q ª C共eC兲 ª
WC共eC兲 ª
冕
冕
ns
Wtotal ª
f C · dt, 共2兲 eC共q兲dq,
q=C共e0兲
where eC共q兲 ª C−1共q兲. Similarly, for the inertance element 共I-兲, we can define its “momentum” p, the constitutive inertance relation, and the energy storage WI by
WI共f I兲 ª
冕
冕
e · dt, 共3兲
I共 f I兲
f I共p兲dp,
p=I共 f 0兲
where f I共p兲 ª I−1共p兲. 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 e0 and f 0 so that WC共eC兲 and WI共f I兲 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 共assuming power bonds point to the elements兲: d W C = e C · f C, dt
eR · f R 艌 0,
共4兲
兺f e . j=1
ij ij
共5兲
In other words, given a set of initial conditions, there exists c 苸 R s.t. for any inputs, and for any time t 艌 0 t
0
s„f i1共兲, . . . , f in 共兲,ei1共兲, . . . ,ein 共兲…d 艌 − c2 . i
˙ W total = − D共f R1, . . . , f Rn ,eR1, . . . ,eRn 兲 + s共f i1, . . . , f in ,ei1, . . . ,ein 兲 R
R
i
i
共6兲 R f e where D共f R1 , . . . , f Rn , eR1 , . . . , eRn 兲 ª 兺nj=1 R j R j is the power disR R sipation in all the resistive elements. Proof. Let the storage function Wtotal be the total energy as suggested in the theorem. Using the constitutive relationship of each C-, I-, and R- element and the continuity of power in the junction structure and the TF and GY elements, it is easy to show that
nR
˙ W total = −
兺e j=1
ni
Rj f Rj
+
兺f e j=1
ij ij
= − D共f R1, . . . , f Rn ,eR1, . . . ,eRn 兲 R
R
+ s共f i1, . . . , f in ,ei1, . . . ,ein 兲 i
i
˙ ⇒W total 艋 s共f i1, . . . , f ik,ei1, . . . ,ein 兲,
共7兲
i
R
k
s共f i1, . . . , f ik,ei1, . . . ,eik兲 ª
according to the constitutive equations (2) and (3) for each C- and I- element. Then, Wtotal is a storage function for the bond graph and it satisfies
D共f R1, . . . , f Rn ,eR1, . . . ,eRn 兲 艌 0,
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 共as signal兲 are also introduced in order to represent a wider class of mechatronic/ control systems. These are represented as full arrows 共→兲 pointing in the direction where the signal is transmitted 共see Fig. 10 for an example兲. Unlike power bonds, the source of the signal bond A in 共A → B兲 is unaffected by the destination of the bond, B. Unfortunately, because of this, signal 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 the ports. We first state the following result for regular bond graphs which will be extended for bond graphs with power scaling elements. 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 ni input (effort or flow) sources, ns C- or I- energy storage elements, and nR R- dissipative elements. Let the sign convention be such that all input bonds point to the bond graph, and power bonds point to each of the C-, I-, and R- elements. Then, with respect to input bonds i1 , i2 , . . . , ni with effort and flow variables ei j , f i j , j = 1 , . . . , ni, the system is passive with respect to the supply rate:
冕
C/I,j
The last inequality is because
d WI = eI · f I . dt
For a resistive element, the constitutive relationship is 共assuming power bonds point to the elements兲 eR ª R共f R兲,
兺W j=1
C共eC兲
p ª I共f I兲 ª
Let the total energies in all the capacitive and inertance elements be given by
i
Journal of Dynamic Systems, Measurement, and Control
R
which is a property of resistive elements. Integrating 共7兲, and using the fact that Wtotal共t兲 艌 0, we obtain the desired passivity property: − W共0兲 = W共t兲 − W共t = 0兲 艋
冕
t
0
s共f i1, . . . , f in ,ei1, . . . ,ein 兲d . i
i
Hence, a system that can be modeled by a regular bond graph 共such as a physical system兲 is passive if all its ideal effort and flow sources 共Se or S f 兲 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 共with power sources embedded兲 with the controller 共the algorithm兲, and with the external environment. The questions being addressed in the subsequent sections are a. b.
how to appropriately represent this power modulation using bond graphs so that the passivity property of the control system can be investigated and how to determine a controller that makes the control system passive and how to represent the “equivalent” passive control system with bond graphs.
3 Power Scaling Transformers (PTFs) and Power Scaling Gyrators (PGYs) 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 nonunity scaling factor that relates the power inputs at the two ports. Specifically, PTF共m , 兲 denotes a power scaling transformer with transformer modulus m and power scaling in Fig. 2. Its effort and flow variables at the two ports are causally related by DECEMBER 2005, Vol. 127 / 635
Fig. 2 Causal relations transformers/gyrators
of
power
scaling
THEOREM 2. If a bond graph with power scaling transformers/ gyrators but no active bonds is singly connected at every nonunity power scaling transformer/gyrator, then with respect to the ni input bonds i1 , i2 , . . . , ini (assuming all the sign conventions of all input bonds correspond to power input into the system when the variables are positive), there exist power scalings 1 , 2 , . . . , ni such that the system is passive with respect to the supply rate: ni
e1 ª me2,
s共f i1, . . . , f in ,ei1, . . . ,ein 兲 ª
f 2 ª 共m兲f 1 , f 1 ª f 2/共m兲.
or e2 ª e1/m,
i
共8兲
As such, m is the kinematic scaling between the two flow variables. Similarly, for PGY共r , 兲, a power scaling gyrator with gyrator modulus r and power scaling in Fig. 2, the relationships between the effort and flow variables are e1 ª rf 2,
e2 ª 共r兲f 1 ,
or f 1 ª 1/共r兲e2,
f 2 ª 共1/r兲e1 .
共9兲
For unity power scaling 共i.e., = 1兲 PTF共m , 兲 and PGY共r , 兲 reduce to regular transformers and gyrators. Notice for both PTF共m , 兲 and PGY共r , 兲, 共e1 f 1兲 = 共e2 f 2兲. Thus, the power at input is scaled by the factor before supplying it to the output. PROPOSITION 1. A power scaling transformer PTF共m , 兲 or a power scaling gyrator PGY共r , 兲 are conserving with respect to the -scaled power input in the sense that s共f 1, f 2,e1,e2兲 ª f 1e1 + 共− f 2e2兲 = 0.
共10兲
Here, the power directions are as shown in Fig. 2. Therefore, a power scaling transformer/gyrator is passive with s共. , . , . , . 兲 in (10) as the supply rate. Proof. Using 共8兲 and 共9兲, the scaled passivity property of individual PTFs and PGYs can be obtained by directly verifying that 䊐 s共f 1 , f 2 , e1 , e2兲 in 共10兲 does indeed vanish identically. DEFINITION 1. A bond graph with power scaling transformers/ gyrators is said to be singly connected at a PTF共m , 兲 or PGY共r , 兲 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 a similar passivity property as a regular bond graph as long as it is singly connected at each nonunity power scaling element.
i
兺 f e . j=1
j ij ij
共11兲
In order words, given a set of initial conditions, there exist c s.t. for any inputs, and for any time t 艌 0,
冕
t
0
s„f i1共兲, . . . , f in 共兲,ei1共兲, . . . ,ein 共兲…d 艌 − c2 . i
i
Proof. 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 km disjoint bond graphs. For bond graph i 艋 km, associate an energy storage Wi to be the sum of the storages of all the I- and C- elements in the bond graph according to Theorem 1. Let si be the supply rate for each disjoint bond graph with respect to which it is passive. Now, recursively reinsert 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 reinserted. Consider the step when the l 艋 kth PTF/PGY is reinserted. Suppose the storage functions of the two bond graphs to be reconnected to ports 1 and 2 of the transformer/gyrator are Tl1, and Tl2, respectively, their supply rates are sl1 and sl2, the dissipation rates are Dl1 艌 0 and Dl2 艌 0, and the power scaling of the transformer/ gyrator is l. Thus, for i = 1 , 2, d Tli = − Dli + sli 艋 sli . dt Notice that according to Theorem 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 Tl = lTl1 + Tl2 and the supply rate to be sl ª lsl1 + sl2. Clearly,
Fig. 3 Example illustrating the proof procedure of Theorem 2. The subbond graphs are reconstituted and the storage functions, supply rates, and dissipation function as subgraphs are combined are scaled and added up.
636 / Vol. 127, DECEMBER 2005
Transactions of the ASME
Table 1 Dimensions in M„ass…, L„ength…, T„ime… of various variables
Fig. 4 A nonpassive bond graph with power scaling transformer that is not singly connected
d d d Tl = l Tl1 + Tl2 = − 共Dl1 + Dl2兲 + 共 · sl1 + sl2兲 dt dt dt 艋 共 · sl1 + sl2兲 ¬ sl ,
d f 1 = 共1 − 1/兲f 1 + u, dt
where u is the input effort. Therefore, the system is neither passive nor stable, when ⬎ 1. The example in Fig. 4 also shows that the singly connectedness condition in Theorem 2 is not necessary, since, for ⬍ 1, the bond graph is passive with respect to the supply rate s ª 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.
4 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 outgoing flow 共QA兲 to the hydraulic actuator and the return flow 共QB兲 from it. Assuming the hydraulic actuator is flow conserving 共e.g., in a double-ended cylinder兲, and neglecting flow forces and valve chamber dynamics, then QL ª QA = QB. A mathematical model of the valve is given by 关16兴 mx¨v = F,
Ps , PA , PB , PL
m
xv
F , Fx ,
Kq
关L3T−1兴
关ML−1T−2兴
关M兴
关L兴
关MLT−2兴
关L2T−1兴
1k , K
x*v = 共1kxv兲
r1
r2
␥1 , ␥2 , ␥
If
关MT−2兴
关MLT−2兴
关M −1L2T兴
关1兴
关1兴
关M −1T2兴
z , z⬘
Fx⬘
B
A
␣
Bw
关LT−1兴
关LT−1兴
关MT−1兴
关L2兴
关MT−1兴
关T−1兴
共12兲
i.e., the combined bond graph is passive with sl as its supply rate and Tl as its 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 共11兲 and the complete reconstituted bond graph is passive with respect to it. 䊐 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. Its dynamics are given by I
QL , QA , QB
共13兲
Fig. 5 A typical four-way directional control valve
Journal of Dynamic Systems, Measurement, and Control
QL共xv, PL兲 =
C dw x 冑Ps − sgn共xv兲PL , v
共14兲
where F is the total longitudinal force experienced by the spool, which can be controlled using an electromechanical/solenoid actuator; xv is the spool displacement; m is the spool inertia; Cd and w are the discharge and area gradient coefficients of the valve; Ps is the supply pressure; and PL is the load pressure 共differential pressure between the actuator ports兲; sgn共·兲 denotes the sign function. Equation 共14兲 is derived by combining the orifice equations for the meter-in and meter-out orifices. It is applicable when sgn共xv兲PL ⬍ Ps, which is the usual scenario. A similar expression can be written for the unusual situation when sgn共xv兲PL 艌 Ps. We now consider the bond graph representations of the valve. To facilitate understanding of the various bond graphs and transformations, the dimensions of the physical quantities and variables in standard M共ass兲, L共ength兲, and T共ime兲 units are tabulated in Table 1. The bond graph model for the valve can be decomposed into the spool dynamics portion and the hydraulics portion. 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 Fig. 6. Notice that the valve displacement xv modulates the outgoing flow from the pressure source to the load, and the return flow from the load to the reservoir, via the two orifices 共modeled using R- elements with parameters modulated by xv兲 关11, pp. 277–278兴. This modulation connects the spool portion and the hydraulic portion of the valve. A simplified model, with the assumptions of incompressible flow and that of the load being flow conserving 共i.e., QA = QB = QL兲, corresponding to Eqs. 共13兲 and 共14兲, is shown in Fig. 7 where PL = PA − PB is the load pressure. From the perspective of control, a valve is a two-port device that interacts with two external environments: the hydraulic load 共via PL and QL兲 and the control system 共via the valve command input兲, and the power supply is simply part of the system. From the model in Fig. 7, it is clear that whenever xv ⫽ 0, it is possible to manipulate the load pressure PL so that the pump pressure source Se : Ps delivers power to the external environment. In other
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.
DECEMBER 2005, Vol. 127 / 637
Fig. 10 Dualized active bond graph representation of four-way directional control valve
Fig. 7 Simplified bond graph of the valve. We wish to develop a control law so that the system „with the energy source included… is passive as it interacts with the load and the command input.
words, as far as the hydraulic environment is concerned, the valve is not passive. Of course, the valve would be passive if Se : Ps were also considered part of the external environment 共Fig. 6兲. 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 关4兴, an alternative representation that is more suitable for bond graph passivity analysis is obtained by first reformulating the flow equation 共14兲 to be
In this new perspective, the goal of passification is to modulate the effort source Se : F with a feedback control so as to make the system appear passive to the external environment.
5
Bond Graph Approach for Passification
Fig. 8 Equivalent electrical circuit for the hydraulic valve equation „15… which is equivalent to Eq. „14…
We now proceed to develop a control law to passify the electrohydraulic valve. This involves several manipulations and transformations of the bond graph in Fig. 9. Please refer to Table 1 for the dimensions of the various variables. Notice that the bond graph in Fig. 9 contains two signal bonds: one associated with the modulating effect of the spool displacement xv on the flow rate, the other associated with the integration of the spool velocity x˙v to obtain the spool displacement xv. 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, of which step 0 is optional. 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. In this step, the “0” and “1” junctions, C- and I- elements, effort and flow variables, and the direction of the causality are exchanged to obtain an equivalent bond graph. This step is optional and it is used so that the resulting power scaling bond graphs to be developed later will have transformer-type instead of gyrator-type causalities. 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 ␥1, ␥2 of the two PTF’s and the modulation factor r2 of the PTF that replaces the “integrator” signal bond are to be determined later. The modulation factor of r1 of the PTF in the xv induced signal bond must be chosen so that Kq = 1kr1␥1 to preserve the relationship between the no-load flow and xv in 共15兲. Notice that if the dualization step in step 0 is omitted, then the active bond on the right-hand side should be replaced by PGY instead of PTF. We prefer to use PTF because they reduce to simple power bonds when both the modulation factor and the power scaling are unity, as opposed to reducing to GY in the case of PGY when the duality transformation is omitted. Whether one dualizes first is simply a stylistic issue. 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 lefthand most “0” junction. The resulting bond graph is shown in Fig. 11. Notice that Fig. 11 is a bond graph with power scaling com-
Fig. 9 Active bond graph representation of four-way directional control valve. 1k is the unit fictitious stiffness „dimension †MT−2‡… associated with the integral relationship between x˙v and xv. xv* ª „1kxv… is the effort variable at the 0 junction.
Fig. 11 Desired power scaling bond graph representation of four-way directional control valve with bonds replaced by PTF/PGY
QL共xv, PL兲 = Kqxv − Kt共xv, PL兲PL ,
共15兲
where Kq = Cdw冑Ps / ⬎ 0 and Kt共xv , PL兲 can be shown to be nonnegative. Thus, we can think of the valve as being a flow source modulated by the spool displacement xv with a no-load flow gain Kq with respect to xv, in parallel with a nonlinear conductance Kt共xv , PL兲 that shunts flow 共Fig. 8兲. The corresponding bond graph model is shown in Fig. 9. Here, the time integration of x˙v to form xv is modeled using a mechanical capacitive element 共e.g., a spring兲 represented by a compliance of 1 / 1k where 1k is the unit stiffness 共i.e., has a value of 1兲. The effort variable at the 0 junction is x*v ª 共1kxv兲 共has the dimension of a force 关MLT−2兴兲. It has the same value as xv关M兴 but with a change in units 共Table 1兲. In Fig. 9, the spool inertia dynamics determines the spool displacement which in turn modulates a flow source with a gain Kq. In contrast, in the bond graph in Fig. 7 using 共14兲, the spool only modulates the resistance values, which has only indirect influence on the energy flow.
638 / Vol. 127, DECEMBER 2005
Transactions of the ASME
ponents 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共F⬘x , PL,xv,QL兲 = 共1k␥1兲F⬘x xv − PLQL . 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 xv and of z are given by x˙v =
1 z + 共F⬘x − r1 PL兲, r2
mz˙ = −
1k x − Bz. ␥ 2r 2 v
共16兲 共17兲
Notice that 共16兲 provides the transformation z that is given by z = x˙v − 共F⬘x − r1 PL兲. r2 Differentiating 共16兲 and utilizing 共17兲, we obtain the spool dynamics necessary to realize the dynamics of the bond graph to be mx¨v = m
Substituting the expression for z, we have d − r1 PL兲. mx¨v = − Bx˙v − x − r1 PL兲 + m 共F⬘ 2 xv − B共F⬘ dt x ␥ 2r 2 共18兲 Comparison between 共18兲 and 共13兲 suggests that the ideal passifying control law should be of the form
= − Bx˙v − Kxv − r1BPL + BF⬘x + m
d 共F⬘ − r1 PL兲, dt x
共19兲
where K = 1k / 共␥2r22兲. 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 关16兴. So, control needs only augment to these. Pressure feedback can be obtained physically by modifying the spool geometry 关4兴. The fourth and last terms in 共19兲 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 共F⬘x , xv兲 and 共PL , QL兲 as the input port variables. Following the proof of Theorem 2, we can choose 1 k␥ 1 2 m W = ␥ 1␥ 2 z 2 + x 2 v 2
共20兲
as the storage function of the system, so that ˙ = − ␥ ␥ Bz2 − K 共x , P 兲P2 + 共共1 ␥ 兲F⬘x − P Q 兲. 共21兲 W 1 2 t v L L k 1 x v L L Hence, the system is passive with respect to the supply rate: svalve共F⬘x , PL,xv,QL兲 ª 共1k␥1兲F⬘x xv − PLQL ,
冏
冏
ˆd d 共F⬘x − r1 PL兲 − 共F⬘x − r1 PL兲 艋 berr , dt dt
Frob = − m · sgn共z兲berr .
1k
d 共F⬘ − r1 PL兲 dt x
The control law requires estimating the derivative of F⬘x − r1 PL. 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 共Fig. 12兲. Assuming that we can estimate the bound for the estimation error,
共23兲
where dˆ / dt共·兲 is the estimate of the derivative of the argument, the passifying control law can be modified to include the term
1k d z 共F⬘x − r1 PL兲 − B − xv . dt r2 ␥2r22
F = − Bx˙v − Kxv + B共F⬘x − r1 PL兲 + m
Fig. 12 Bond graph of passified valve with robustness modification and estimation error
共22兲
where 共1k␥1兲F⬘x xv represents the power at the command port, and PLQL represents the power at the hydraulic port. The second term in 共21兲 represents the valve intrinsic energy dissipation due to the shunt conductance in Kt共xv , PL兲 in 共15兲. The first term in 共21兲 is the dissipation, which is in fact an artifact of the proposed bond graph. This term is generally desired to be small. Step 4. Adding robustness. Journal of Dynamic Systems, Measurement, and Control
共24兲
This ensures that Frobz + Error· z 艋 0 for any estimation error Error共·兲 关the signal inside 兩·兩 in 共23兲兴 satisfying its assumed bound. The penalty for using a conservative error bound in the robustness control term would be a large additional dissipation term −␥1␥2mberr兩z兩 in 共21兲. 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 共r1 , ␥1 , r2 , ␥2 , B兲. However, valve orifice relationship dictates that 共1k␥1兲r1 = Kq, the no load flow gain. Let ␥ ⬎ 0, A ⬎ 0 be two constants and then define r1 ª
A , ␣
r2 ª 1,
␥1 =
Kq , r1
1 ␥2 = , ␥B
共25兲
as well as a transformed input Fx ª ␣ · F⬘x so that F⬘x − r1 PL =
1 共Fx − APL兲. ␣
共26兲
Then, the passifying control law 共19兲, with the robustness term in 共24兲, recovers exactly the active passifying control law in 关4兴 by setting ␣ = B. In this case, the passifying control 共19兲 is parameterized by 共r1 = A / ␣ , ␥ , B兲. ␣ only plays the role of input scaling as in 共26兲 but does not alter the passifying control for a given A / ␣. The possibility of arbitrary input scaling using ␣ ⫽ B is unknown in 关4兴 without using the bond graph approach. Examination of the target bond graph dynamics 共16兲 and 共17兲 shows that r2 does not play a role in terms of xv dynamics. This, together with the constraint Kq = 1k␥1r1, means that the passifying control law suggested by the bond graph in Fig. 11 is completely parameterized by 共r1 = A / ␣ , ␥ , B兲. The closed loop transfer function of the valve passified using the parameters in 共25兲 and the input scaling 共26兲 is DECEMBER 2005, Vol. 127 / 639
xv共s兲 = =
s + B/m 关F⬘共s兲 − r1 PL共s兲兴 关s共s + B/m兲 + ␥B/m兴 x s + B/m 关Fx共s兲 − APL共s兲兴. ␣关s共s + B/m兲 + ␥B/m兴
共27兲
If we set both poles at s = −Bw, where Bw signifies the bandwidth of the passified valve, we need B = 2mBw and ␥ = Bw / 2. We get the spool dynamics and the output flow equation: xv共s兲 =
s + 2Bw 关Fx共s兲 − APL共s兲兴, ␣共s + Bw兲2
共28兲
QL共xv, PL兲 = Kqxv − Kt共xv, PL兲PL .
共15⬘兲
Fig. 13 Alternate bond graph structures for passification
If the bandwidth of operation is well below Bw, the passified valve can be approximated by its static gain: xv ⬇
Y共s兲 =
2 共Fx − APL兲, ␣Bw
冢 冣冢
1
x˙v
This static approximation has facilitated the development of the first successful passive bilateral teleoperation of a hydraulic actuator 关5,6兴. When the static approximation holds, from 共21兲, the dissipation in the valve is given by
冋冉 冊
册
Kq F xx v − P LQ L ; A
mz˙⬘
˙ ⬇ − ␣Bw Kq x2 − K 共x , P 兲P2 + W t v L L 2 A v
冋冉 冊
˙ If
xv共s兲 =
0
1
0
= −K −B −1 0 1 0
册
Here, the second term represents the actual energy loss due to the shunt conductance in the valve 共15兲; the last term is the supply rate consisting of the control input power 共Kq / A兲Fxxv and the hydraulic output power PLQL. The first term represents energy dissipation which is a consequence of our passification algorithm. In particular, for the same valve opening xv, the dissipation is proportional to ␣Bw. In the teleoperation control in 关5,6兴, this appears as an extra damping in the haptic property of the controlled system and adversely affects the way that a human perceives and distinguishes the external environment. If ␣ = B = 2Bwm as in 关4兴 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 ␣ in the bond graph approach, the passificationinduced dissipation can be made arbitrarily small by simply adjusting ␣.
z⬘
+
0
共Fx − APL兲, 共30兲
0
冋
mI f 共KI f + 1兲s共s + B/m兲 + 共KI f + 1兲 1 ␣共KI f + 1兲s mI f s共s + B/m兲 + 共KI f + 1兲
册
Generalization
We already saw above that the use of ␣ effectively removes the apparent trade-off between bandwidth and dissipation. The bond graph approach also offers potential new ways to passify the valve. For example, target bond graph structure alternates to Fig. 11, such as Fig. 13, can be used. Here, a general admittance Y共s兲 as well as an additional input F1 are attached to the “0” node, and a general impedance Z共s兲 and an input Fx 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:
共31兲
Both poles of the second-order component inside the 关·兴 can be set at s = −Bw where Bw is the desired bandwidth. The low-frequency valve dynamics can be approximated by the integrator dynamics 共take s → 0兲: x˙v ⬇
1 共Fx − APL兲. ␣共KI f + 1兲
The integrator valve dynamics are attractive because command is now related to dQL / 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 = r2z⬘ ⬇ 0. This can be seen by using the storage function: Wmod ª ␥1␥2
冉
冊 冋冉 冊
1 k␥ 1 2 If 2 m 2 x , + z + 2 v 2 2
2 2 ˙ W mod = − ␥1␥2Bz − Kt共xv, PL兲PL +
⬇ − Kt共xv, PL兲PL2 +
640 / Vol. 127, DECEMBER 2005
冣冢 冣 冢 冣 1/␣
xv
⫻关Fx共s兲 − APL共s兲兴.
Kq F xx v − P LQ L . A 共29兲
6
F1 = 0.
where we have used the substitution z⬘ = z / r2, K = 1k / 共␥2r22兲, r1 = A / ␣, and Fx = ␣F⬘x . This has the transfer function:
1
using z ⬇ − 2 Bwxv during low frequency operation,
Z共s兲 = 0,
The bond graph for Y共s兲 is shown in Fig. 13. The valve dynamics become
z ⬇ − ␥ x v = − 2 B wx v .
˙ = 2␣ Kq z2 − K 共x , P 兲P2 + W t v L L ABw
If + B, s
冋冉 冊
Kq F xx v − P LQ L A
册
Kq F xx v − P LQ L . A
共32兲
册 共33兲
Moreover, since the energy dissipation due to implementing the robustness term 共24兲 is proportional to 兩z兩, this will also vanish asymptotically. Therefore, the penalty for poorly estimating 共d / dt兲共F⬘x − r1 PL兲 is also reduced.
7
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 Transactions of the ASME
analysis. It is shown that singly connected power scaling bond graphs are passive with respect to an appropriately scaled power input. 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 electro-hydraulic 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 to obtaining a tighter condition to determine when a power scaling bond graph is passive, and thus increasing the applicability of the proposed approach.
Acknowledgment Research supported by the National Science Foundation under Grant No. CMS-0085230.
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