Modelling of Integrated Systems for Modular Linearisation-Based Analysis And Design

Transcription

1 Modelling of Integated Systems fo Modula ineaisation-based Analysis And Design D.J. eith, W.E. eithead Depatment of Electonic & Electical Engineeing, Univesity of Stathclyde, 5 Geoge St., Glasgo G QE, U.K. Tel , Fax , . doug@icu.stath.ac.uk Keyods Integated systems, lineaisation, modula analysis Abstact The selection of an appopiate epesentation fo lineaisation-based analysis and design of integated systems is consideed. It is shon that, in contast to conventional lineaisation-based epesentations, a velocity-based epesentation suppots the modula analysis and design methodologies equied ith complex integated systems.. Intoduction In esponse to inceasingly stingent pefomance equiements, the tend in a ide ange of engineeing systems is toads tighte integation of the elements constituting the system. Whilst this fequently leads to inceased dynamic inteaction beteen sub-systems, the design of a complex system in an efficient and flexible manne necessitates a modula methodology heeby each sub-system has a ell-defined inteface to the est of the system hich is insensitive to the implementation details of the system. Such an appoach enables the detailed design and implementation of each sub-system to be caied out sepaately and this is paticulaly impotant in pojects hee sub-contactos ae involved. Thee is, of couse, a coesponding equiement fo a epesentation hich suppots the analysis and design of systems in a modula fashion. Depending on the task at hand, a numbe of diffeent epesentations ae typically utilised fo the analysis and design of complex systems; fo example, Bond Gaphs, dinay Diffeential Equations and Tansfe Functions. Each epesentation possesses paticula advantages and disadvantages hich make it moe, o less, suitable fo a paticula pupose. Taditionally, lineaisation-based epesentations ae often employed fo the analysis and design of nonlinea systems. The system is appoximated by a suitable linea system hich, in the vicinity of an equilibium opeating point, exhibits simila dynamic chaacteistics (typically, the seies expansion lineaisation about the equilibium opeating point. Whilst conventional lineaisation-based analysis is only valid locally to a specific equilibium opeating point, it has the consideable advantage that it maintains continuity ith established linea analysis techniques fo hich a substantial body of expeience has been accumulated. In ode to detemine appopiate linea appoximations to a paticula sub-system, knoledge of the equilibium opeating points is equied. Hoeve, specifying a pioi the elationship beteen the equilibium inputs and outputs of a sub-system imposes, in geneal, a vey stong estiction on the chaacteistics of the sub-system. Fo example, conside the nonlinea system x F(x,, y G(x,. The functions, F(, and G(,, ae not independent since, F(x o, o, G(x o, o y o must be jointly satisfied fo all equilibium input-output pais (x o, o. Indeed, hen F(, is invetible so that the equilibium state, x o, is detemined by o, it follos that the output function, G(,, is, essentially, completely specified by the equilibium input-output elationship and the choice of F(,. Consequently, in pactice, a flexible equilibium specification is equied. Hoeve, since the equilibium opeating points ae not those of the isolated sub-system but athe those of the sub-system hen it is embedded in the oveall system, the equilibium opeating points of a sub-system ae, in geneal, stongly influenced by the chaacteistics of the oveall integated system and may change consideably as a esult of even elatively small changes in othe sub-systems. Hence, conventional lineaisation-based epesentations do not eadily suppot modula analysis and design appoaches. Whilst, in the context of integated systems, the foegoing is pehaps the pimay deficiency of conventional lineaisation-based epesentations, it should be noted that thee ae also a numbe of othe difficulties ith such techniques. Fistly, the epesentations ae only accuate in the vicinity of an equilibium opeating point hilst the equiement is usually to design a system hich functions ell not only hen opeating in the vicinity of a single equilibium point but also duing tansitions beteen equilibium opeating points and peiods of sustained non-equilibium opeation. Conventionally, this equiement is addessed by employing extensive simulation studies to iteatively efine the design, but this quickly becomes extemely time-consuming and inefficient fo any but the simplest nonlinea systems. Thee is, theefoe, a consideable incentive to diectly incopoate, into the analytical pat of the design pocedue, knoledge of the plant dynamics duing tansitions beteen equilibium opeating points and duing sustained non-equilibium opeation. Secondly, a numbe of distinct linea epesentations ae typically employed duing analysis and design (eith & eithead 998a hich ae not equivalent and hich can make it difficult to incopoate insight obtained fom the analysis into the design pocedue. Thidly, and at a moe pactical level, the detemination of the equilibium opeating point is time-consuming and highly non-tivial fo a complex

2 nonlinea system. Similaly, the numeical diffeentiation associated ith conventional numeical lineaisation about an equilibium point is an undesiable ill-conditioned opeation. The velocity-based epesentation ecently poposed by eith & eithead (998a igoously genealises and extends the conventional seies expansion lineaisation at an equilibium opeating point and associates a linea system ith evey opeating point, not just equilibium opeating points. An altenative desciption of a nonlinea system in tems of a family of linea systems is theeby established; namely, the velocity-based lineaisation family. It is emphasised that the velocity-based fomulation involves no loss of infomation and, in paticula, does not involve eithe an inheent slo vaiation estiction no is it confined to equilibium opeating points alone but instead encompasses evey opeating point, including those fa fom equilibium. The velocity-based appoach theeby elaxes the estiction to nea equilibium opeation hilst maintaining the continuity ith linea methods hich is a pinciple advantage of the conventional lineaisation-based analysis techniques. The velocity-based epesentation theefoe esolves many of the deficiencies of conventional lineaisation-based epesentations. Hoeve, the liteatue on the velocity-based epesentation is confined to monolithic systems fo hich thee is no equiement to conside a decomposition into component sub-systems. The aim of this pape is, theefoe, to investigate the application of the velocity-based epesentation to integated systems and, in paticula, conside the suppot, if any, povided by this epesentation fo modula analysis and design.. Velocity-based epesentation Befoe consideing integated systems, the velocitybased analysis and design epesentation is biefly summaised. Conside a nonlinea system x F(x,, y G(x, ( hee F(, and G(, ae diffeentiable nonlinea functions ith ipschit continuous fist deivatives and R m denotes the input to the plant, y R p the output and x R n the states. The set of equilibium opeating points of the nonlinea system, (, consists of those points, (x o, o, fo hich F(x o, o ( and the equilibium output is y o G(x o, o (3 et Φ:R n R m denote the space consisting of the union of the states, x, ith the inputs,. The set of equilibium opeating points of the nonlinea system, (, foms a locus of points, (x o, o, in Φ and the esponse of the system to a geneal time-vaying input, (t, is depicted by a tajectoy in Φ. The solution x to the linea system (the velocitybased lineaisation "x, " x F(x, F(x, (4 "y x G(x, G(x, (5 appoximates the solution x to the nonlinea system locally to the opeating point (x,. Since a linea system (4-(5 is associated ith evey opeating point of the nonlinea system, thee is a family of velocity-based lineaisations associated ith the nonlinea system. Whilst the solution to a single velocity-based lineaisation is only a local appoximation to the solution of the nonlinea system, the solutions to the membes of this family can be pieced togethe to obtain an abitaily accuately global appoximation to the solution of the nonlinea system. The elationship beteen the nonlinea system and its velocitybased lineaisation, (4-(5, is diect. Diffeentiating (, an altenative epesentation of the nonlinea system is x, x F(x, F(x, (6 y x G(x, G(x, (7 Dynamically, the velocity-fom (6-(7, ith appopiate initial conditions, and ( ae equivalent (have the same solution, x. Clealy, the velocity-based lineaisation, (4- (5, is simply the foen fom of (6-(7 at the opeating point, (x,. Hence, numeical diffeentiation is not equied in ode to lineaise a system in velocity-based fom; instead, the velocity-based lineaisation is obtained by simply feeing the velocity fom of the nonlinea system. The velocity-based fomulation, (6-(7, is of higheode than the diect fomulation, (. Hoeve, efomulate (, as x Ax B f(ρ, y Cx D g(ρ (8 hee A, B, C, D ae appopiately dimensioned constant matices, f( and g( ae nonlinea functions and ρ(x, R q, q mn, embodies the nonlinea dependence of the dynamics on the state and input ith x ρ, ρ functions of ρ alone. Tivially, this efomulation can alays be achieved by letting ρ [x T T ] T, in hich case qmn. The nonlineaity of the system is, hoeve, fequently dependent on only a subset of the elements of the state and input, in hich case the dimension, q, of the scheduling vaiable ρ is less than mn. Assume, ithout loss of geneality, that x ρ and ρ ae constant. The velocity-fom of the system, (8, is x (9 (Af(ρ x ρ (Bf(ρ ρ ( y (Cg(ρ x ρ (Dg(ρ ρ ( hich, as ( and ( depend only on, ρ and, may be efomulated, equivalently, as ρ x ρ ρ ( (Af(ρ x ρ (Bf(ρ ρ (3 y (Cg(ρ x ρ (Dg(ρ ρ (4 Since the dimension of the scheduling vaiable ρ is, typically, much loe than that of x, the ode of (-(4 is usually only slightly highe than the ode of (. Thee exists a igoous, and diect, elationship beteen the dynamic chaacteistics of a nonlinea system and those of its velocity-based lineaisation family. It is emphasised that the velocity-based lineaisation family embodies the entie dynamics of the nonlinea system, (,

3 ith no loss of infomation and povides an altenative epesentation of the nonlinea system. Thee is no estiction to nea equilibium opeation no any slo vaiation equiement. Whilst the velocity-based epesentation is equivalent to the diect epesentation, (, in the sense that they each embody the entie dynamics of the nonlinea system, they ae not necessaily equivalent ith espect to othe consideations. In paticula, the diect elationship beteen the velocity-fom of the nonlinea system and the velocity-based lineaisation family and the lineaity of the membes of the latte family povides continuity ith established linea theoy hich, fo example, facilitates analysis (eith & eithead 998a and design (eith & eithead 998b,c. With egad to design, it is noted that the velocitybased epesentation povides diect suppot fo divide and conque design appoaches, such as the gain-scheduling design methodology, heeby the design of a nonlinea system is decomposed into the design of an associated family of linea systems (eith & eithead 998b,c. Fo example, since a velocity-based lineaisation family is associated ith a nonlinea plant, a coesponding linea contolle family can be obtained by designing a linea contolle fo each membe of the plant family. A nonlinea contolle may then be detemined fo hich the velocity-based lineaisation family is the designed linea contolle family. This appoach esolves many of the deficiencies of the conventional gain-scheduling appoach including the estiction to an excessively small neighbouhood of the equilibium opeating points. By alloing infomation about the plant dynamics at nonequilibium opeating points to be diectly incopoated into the contolle design, both sustained non-equilibium opeation and dynamic tansitions beteen equilibium opeating points, including those hich take the system fa fom equilibium, can be accommodated. In paticula, the velocity-based gain-scheduling appoach can be employed to detemine a dynamic invesion contolle hich achieves global linea dynamics hen combined ith the nonlinea plant (eith & eithead 998c. f couse, the velocity-based gain-scheduling appoach is quite geneal and also diectly suppots the design of feedback configuations fo hich the closed-loop dynamics ae nonlinea. 3. Integated systems The velocity-based epesentation esolves many of the deficiencies of conventional lineaisation-based analysis and design techniques. Hoeve, the existing liteatue on the velocity-based epesentation is confined to monolithic systems fo hich thee is no equiement to conside a decomposition into component sub-systems. With egad to integated systems, the velocity-based epesentation, in contast to conventional lineaisationbased epesentations, is not esticted to nea equilibium opeation and does not equie the equilibium opeating point of a system to be detemined befoe lineaisationbased analysis is possible. Rathe, application of velocitybased analysis and design techniques only involves the much eake equiement that the lagest opeating envelope of a system is knon. As noted peviously, it is non-tivial to detemine the equilibium opeating point of a sub-system hen it is embedded in the oveall system and, futhemoe, the equilibium opeating point is, in geneal, stongly influenced by the chaacteistics of the oveall integated system. Since the equiement to detemine an equilibium opeating point is elaxed in the velocity-based fameok, this fameok esolves one of the pincipal difficulties, in the context of integated systems, ith conventional lineaisation-based techniques. f couse, hilst this is necessay in ode to de-couple the analysis and design of the sub-systems, it is not sufficient to suppot modula analysis and design. It is, in addition, also equied that the analysis and design esults obtained ith a specific sub-system can be integated in a diect and tanspaent manne ith those obtained fo othe subsystems. In ode to investigate the integation of analysis and design esults obtained fo diffeent sub-systems, it is sufficient to conside the seies, paallel and feedback combination of sub-systems since these ae the pincipal classes of inteconnection in idespead use. 3. Seies combination Conside the nonlinea system F ( x,,, y G( x,, (5 fo hich the velocity-based fom is x x F (x,, F (x,, F (x,, (6 y x G (x,, G (x,, G (x,, and the nonlinea system F ( x,,, y G( x,, (7 fo hich the velocity-based fom is x x F (x,, F (x,, F (x,, (8 y x G (x,, G (x,, G (x,, The systems, (5 and (7, ae cascaded togethe by setting y. The esulting system is F( x,,, y G( x,, (9 hee x x y y N M x Q P N M Q P,,, F x F x G x G x G x N M ( (,, (,, F x G x Q P (,, (,,, (,, (,, (,, fo hich the velocity-based fom is

4 N M Q P N M Q P N M N M N M x F ( x,, F ( x,, G ( x,, F ( x,, x x F ( x,, F ( x,, G ( x,, F ( x,, Q P Q P N M Q P F ( x,, G ( x,, F ( x,, x x N M Q P y y G ( x,, G ( x,, G ( x,, G ( x,, G ( x,, Q P N M Q P N M Q P G x G x G x (,, (,, (,, y G( x,, Evidently, ( is just the system obtained hen the systems, (6 and (8, ae cascaded togethe. It follos that the velocity-based fom of a system consisting of to cascaded sub-systems is identical to the system obtained by cascading togethe the velocity-based foms of the to sub-systems. ( 3. Paallel combination The systems, (5 and (7, ae combined in paallel by setting. The esulting system is F( x,,, y G( x,, ( hee x y x y x,,, y F( x,, F ( x,, G x G x F ( x,,, (,, (,, G ( x,, (3 fo hich the velocity-based fom is y x F ( x,, F ( x,, x G ( x,, F ( x,, x F ( x,, F ( x,, G ( x,, y x G( x,, y F ( x,, F ( x,, x G ( x,, G( x,, G ( x,, G( x,, G ( x,, (4 Evidently, ( is just the system obtained hen the systems, (6 and (8, ae combined in paallel. Hence, the velocity-based fom of a system consisting of the paallel combination of to sub-systems is identical to the system obtained by combining in paallel the velocitybased foms of the to sub-systems. 3.3 Feedback combination Conside the nonlinea system ith inputs, and, F( x,,, y G( x,, (5 fo hich the coesponding velocity-based fom is x x F(x,, F(x,, F(x,, (6 y x G(x,, G(x,, G(x,, Assuming that yg(x,,y (7 has a suitable solution yn(x, (8 the system, (5, is enclosed in a feedback loop by setting y. The esulting closed-loop system is M( x,, y N( x, (9 ith M( x, F( x,, N( x, (3 The velocity-based fom of (9 is x x M(x, M(x, (3 y x N(x, N(x, Combining (7 and (8 N( x, G( x,, N( x, (3 Hence, M( x, F( x,, N( x, F( x,, N( x, N( x, x x x M( x, F( x,, N( x, F( x,, N( x, N( x, (33 N( x, G( x,, N( x, G( x,, N( x, N( x, x x x N( x, G( x,, N( x, G( x,, N( x, N( x, and, by substituting (33 into(3, the closed-loop system, (9, can be diectly efomulated as x x F(x,, F(x,, F(x,, (34 y x G(x,, G(x,, G(x,, y N(x, Since N(x, satisfies (3, it is clea that (34 is the system obtained hen the system, (6, is enclosed in a feedback loop by setting y. Consequently, the velocity-based fom of the closed-loop system is identical to the system obtained by enclosing the velocity-based fom of the openloop system in a feedback loop. 3.4 Example Conside to nonlinea dynamic systems F ( x, (35 F ( x, (36 hee 3 F ( x,. x x., F ( x,. x. tanh( x 65.. (37 The unfoced equilibium state of (35 is 3.93 and that of (36 is.. The coesponding equilibium lineaisations ae

5 δ δx δ (38 x" δx 393., δ. and δ -. δx. δ (39 x" δx., δ. Hoeve, suppose that the systems ae coupled togethe ith equal to x and equal to x. The equilibium value of (x, x is no (-.78, -.8 and the coesponding equilibium lineaisations ae δ δx δ N M δx (4 x" δx, x" -.78 δx -.8 In compaison, the dynamics obtained by coupling togethe the equilibium lineaisations (38 and (39, δ δ x δ δx (4 N M x" δ., " x 393 x δx. ae quite diffeent fom the linea dynamics, (4; fo example, the eigenvalues of (4 ae (4.846, hilst those of (4 ae (-.968±j.684. This aises because the unfoced equilibium opeating point changes hen the systems ae coupled togethe. In ode to apply conventional lineaisation-based analysis to a paticula sub-system, it is, theefoe, necessay to detemine the equilibium opeating point of the complete coupled system. This is highly non-tivial fo complex nonlinea systems (unlike the simple example pesented hee. Futhemoe, the equilibium opeating point is, in geneal, influenced by the chaacteistics of the evey sub-system. Consequently, conventional lineaisation-based analysis techniques do not eadily suppot a modula methodology. The nonlinea systems, (35 and (36, may be efomulated as the velocity-based foms,. 3x. d i (4 d i b g (43,.. tanh( x 33.. The velocity-based lineaisation families of (35 and (36 ae defined, espectively, by the foen foms of (4 and (43. The velocity-fom of the nonlinea system obtained hen the to systems ae coupled togethe ith equal to x and equal to x is x. 3x x... tanh( x (44 and the membes of the velocity-based lineaisation family of the coupled system consist of the foen foms of (44 (o, equivalently, ae obtained by coupling the elevant membes of the velocity-based lineaisation families of the individual nonlinea systems (35 and (36. The velocitybased lineaisation associated ith an opeating point indicates the dynamics of the nonlinea system at that opeating point. The impulse esponses of the velocitybased lineaisations associated ith a numbe of opeating points of coupled nonlinea systems ae depicted in figue : the vaiation of the dynamic chaacteistics ith the opeating point is clealy evident. The velocity-based epesentation does not equie an equilibium opeating point to be detemined in ode to analyse a system; athe, application of velocity-based analysis and design techniques only involves the much eake equiement that the lagest opeating envelope of a system is knon. In contast to conventional lineaisationbased epesentations, the velocity-based epesentation theefoe suppots modula lineaisation-based analysis/design heeby the analysis/design of a subsystem is, as fa as possible, insensitive to the intenal details of the othe sub-systems constituting an integated system. Futhemoe, the foegoing analysis establishes that the velocity-based lineaisation families of the seies, paallel and feedback combination of to nonlinea systems ae identical to the seies, paallel and feedback combination of the membes of the velocity-based lineaisation families of the individual nonlinea systems. Hence, in addition to de-coupling the analysis/design of sub-systems, analysis and design esults utilising the velocity-based epesentation of a specific sub-system can be integated in a diect and tanspaent manne ith those obtained fo othe sub-systems. 4. Conclusions It is shon that the velocity-based epesentation suppots modula lineaisation-based analysis and design of integated systems In paticula, and in contast to conventional lineaisation-based epesentations, the velocity-based epesentation does not equie an equilibium opeating point to be detemined in ode to analyse a system; athe, application of velocity-based analysis and design techniques only involves the much eake equiement that the lagest opeating envelope of a system is knon. Timming, hich is highly non-tivial fo complex nonlinea systems, is not equied. does not equie numeical diffeentiation; athe the velocity-based lineaisation is obtained by simply feeing the velocity fom of the nonlinea system. is not confined to nea equilibium opeation but athe accommodates both tansitions beteen equilibium opeating points and sustained non-equilibium opeation. povides a unified fameok fo analysis and design hich employs a single lineaisation, namely the velocity-based lineaisation. Futhemoe, the velocity-based epesentations of the seies, paallel and feedback combination of to nonlinea systems ae identical to the seies, paallel and feedback combination of the velocity-based epesentations of the individual nonlinea systems. Hence, in addition to decoupling the analysis/design of sub-systems, analysis and design esults utilising the velocity-based epesentation of

6 a specific sub-system can be integated in a diect and tanspaent manne ith those obtained fo othe subsystems. The velocity-based epesentation theefoe esolves many of the difficulties associated ith conventional lineaisation-based epesentations and povides diect suppot fo the modula analysis and design methodologies equied ith complex integated systems. Acknoledgement D.J.eith gatefully acknoledges the suppot povided by the Royal Society fo the ok pesented. Refeences EITH, D.J., EITHEAD, W.E., 998a, Gain-Scheduled & Nonlinea & Systems: Dynamic Analysis by Velocity- Based ineaisation Families. Int. J. Contol, 7, pp89-37; 998b, Gain-Scheduled Contolle Design: An Analytic Fameok Diectly Incopoating Non- Equilibium Plant Dynamics. ibid, 7, pp49-69; 998c, Input-utput ineaisation by Velocity-based Gain- Scheduling. ibid, in pess.; 998d, Analytic Fameok fo Blended Multiple Model Systems Using inea ocal Models. ibid, in pess. x (x,x (-.4, (x,x (, (x,x (5, x Figue Impulse esponses of the velocity-based lineaisations associated ith a numbe of opeating points. It should be noted that the dynamics ae stongly dependent on the value of x but quite insensitive to the value of x. Responses ae, theefoe, only depicted fo opeating points at hich x is equal to eo. p