Gain-Scheduled Controller Design: An Analytic Framework Directly Incorporating Non-Equilibrium Plant Dynamics

Transcription

1 Gain-Sheduled Contolle Design: An Analyti Fameok Dietly Inopoating Non-Equilibium Plant Dynamis D.J.Leith W.E.Leithead Abstat Depatment of Eletoni & Eletial Engineeing, Univesity of Stathlyde, GLASGOW G QE, U.K. In this pape, a veloity-based lineaisation fameok is employed to develop a novel igoous appoah to gain-sheduling design. The poposed appoah enables knoledge onening the plant dynamis at non-equilibium opeating points to be inopoated dietly into the ontolle design. Sine the veloity-based lineaisation fameok suppots the analysis of the tansient esponse, pefomane onsideations an be aommodated. The appoah etains ontinuity ith linea methods, hih is ental to the eisting onventional gain-sheduling methodology, and, sine a single type of lineaisation is employed thoughout, the design poedue is both staightfoad and oneptually appealing.. Intodution Whilst nonlinea dynami systems ae idespead, the analysis and design of suh systems emains elatively diffiult. In ontast, tehniques fo the analysis and design of linea time-invaiant systems ae athe bette developed even though systems ith genuinely linea time-invaiant dynamis do not, in eality, eist. It is, theefoe, attative to adopt a divide and onque philosophy heeby the design of a nonlinea system is deomposed into the design of a family of linea time-invaiant systems. This type of stategy foms the basis of one of the most idely, and suessfully, applied tehniques fo the design of nonlinea ontolles; namely, gain-sheduling. Gain-sheduled ontolles ae linked by the design appoah employed, heeby a nonlinea ontolle is onstuted by intepolating, in some manne, beteen the membes of a family of linea time-invaiant ontolles. In the onventional, and most ommon, gain-sheduling design appoah (see, fo eample, Astom & Wittenmak 989, Hyde & Glove 993), eah linea ontolle is typially assoiated ith a speifi equilibium opeating point of the plant and is designed to ensue that, loally to the equilibium opeating point, the pefomane equiements ae met. (The eistene of a family of equilibium opeating points, hih spans the envelope of plant opeation, is ental to most gain-sheduling aangements and it is not suffiient to estit onsideation to a single, isolated, equilibium opeating point). By employing a fist-ode seies epansion appoimation hih, loally to the equilibium opeating point, has simila stability popeties to the plant, linea tehniques may be applied to this loal design task. Hoeve, the equiement is usually fo a ontolle hih funtions ell not only hen opeating in the viinity 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 etensive simulation studies to iteatively efine the gain-sheduled design, but this quikly beomes etemely time-onsuming and ineffiient fo any but the simplest nonlinea plants. Thee is, theefoe, a onsideable inentive to dietly inopoate, into the analytial pat of the design poedue, knoledge of the plant dynamis duing tansitions beteen equilibium opeating points and duing sustained non-equilibium opeation. In Leith & Leithead (997b,), a fameok is poposed fo the analysis of gain-sheduled and nonlinea systems hih assoiates a family of veloity-based lineaisations ith a nonlinea system. Eah opeating point of the nonlinea system, inluding opeating points fa fom equilibium, has an assoiated membe of the veloity-based lineaisation family hih desibes the dynami haateistis in the viinity of that opeating point. Hene, in ontast to the onventional seies epansion lineaisation about an equilibium opeating point, the veloity-based lineaisation family indiates the plant dynamis not only in the viinity of a single equilibium opeating point but also duing tansitions beteen equilibium opeating points and hen opeating fa fom equilibium. The pupose of this pape is to investigate the diet eploitation of infomation onening the plant dynamis at non-equilibium points in gain-sheduling design by employing veloity-based

2 lineaisation families. The pape is oganised as follos. In setion, the eisting gain-sheduling design appoahes ae evieed and, in setion 3, the veloity-based lineaisation families ae desibed. A fameok employing veloity-based lineaisations fo the analysis and design of gainsheduled ontolles is developed in setion 4 and speialised to the lass of plants satisfying an etended loal linea equivalene ondition in setion 5. The onlusions ae summaised in setion 6.. Conventional gain-sheduling design Conside the nonlinea plant ith dynamis, F(, ), y G(, ) () hee F(, ) and G(, ) ae ontinuous ith Lipshitz ontinuous fist deivatives, R m denotes the input to the plant, y R p the output and R n the state. When neessay, assume [ F F F ( F) n- F] has ank n,. The set of equilibium opeating points of the nonlinea plant, (), onsists of those points, ( o, y o, o ), fo hih F( o, o ), y o G( o, o ) () Let Φ:R n R m denote the spae onsisting of the union of the state,, ith the input,. The set of equilibium opeating points of the nonlinea plant, (), foms a lous of points, ( o, o ), in Φ and the esponse of the plant to a geneal time-vaying input, (t), is depited by a tajetoy in Φ. The gain-sheduled design appoah onstuts a nonlinea ontolle, ith etain equied dynami popeties, by intepolating, in some sense, beteen the membes of an appopiate family of linea time-invaiant ontolles. The onventional gain-sheduling design appoah may be applied dietly to a boad ange of nonlinea plants and the design poedue typially involves the folloing steps (see, fo eample, Astom & Wittenmak 989 setion 9.5, Shamma & Athans 99, Hyde & Glove 993, Leith & Leithead 996).. The equilibium opeating points of the plant ae paameteised by an appopiate quantity, ρ, hih may involve the plant input, output and/o state.. The plant dynamis, (), ae appoimated, loally to a speifi equilibium opeating point, ( o, o,y o ), at hih ρ equals ρ o, by the seies epansion lineaisation, δ F( o (ρ o ), o (ρ o ))δ F( o (ρ o ), o (ρ o ))δ (3) δ y G( o (ρ o ), o (ρ o ))δ G( o (ρ o ), o (ρ o ))δ (4) δ - o (ρ o ), y δ y y o (ρ o ),δ - o (ρ o ) (5) 3. Fo a suitable ontolle input, e, ith equilibium value, e o (ρ o ), a linea time-invaiant ontolle is designed, δ z A(ρ o )δz B(ρ o )δe (6) δ C(ρ o )δz D(ρ o )δe (7) δe e - e o (ρ o ), δ o (ρ o ) (8) hih ensues appopiate losed-loop pefomane hen employed ith the plant lineaisation, (3)-(5). It should be noted that ρ o is assumed to be onstant hen designing this linea ontolle. 4. Repeat steps and 3 as equied fo a family of equilibium opeating points, ensuing that the linea ontolle designs have ompatible stutues; fo eample, hen a smoothly gain-sheduled ontolle is equied, the linea ontolle designs ae seleted to pemit smooth intepolation, in some appopiate manne, beteen the designs. A family of linea time-invaiant ontolles is obtained oesponding to the family of equilibium opeating points; both the ontolle family and the equilibium opeating points ae paameteised by ρ. 5. Implement the ontolle input and output tansfomations, (8). Typially, the ontolle input, ey-y ef, is zeo in equilibium (that is, e o (ρ o ) and δe e) and eithe the plant ehibits pue integal ation, so that o (ρ) is identially zeo, o eah linea ontolle ontains integal ation hih impliitly geneates o (ρ) though the ation of the feedbak loop (see, fo eample, Astom & Wittenmak 989 setion 9.5, Shamma & Athans 99, Hyde & Glove 993, Leith & Leithead 996). Altenatively, the ontolle output tansfomation may be implemented by epliitly alulating the equilibium ontolle output as a funtion of ρ (see, fo eample, Rugh 99, Shamma & Athans 99). Hoeve, the latte appoah may involve athe omple alulations hih ae sensitive to modelling eos and, onsequently, seems to be lagely of theoetial inteest (Hyde & Glove 99). 6. Substitute ρ (o some elated quantity) fo ρ o in the family of loal linea ontolles, (6)-(8), to obtain a nonlinea ontolle. It is noted that the sheduling vaiable need not be ontinuous; fo

3 eample, it may be piee-ise onstant, oesponding to sithing beteen the membes of the family of loal linea ontolles. Typially, the seletion of an appopiate sheduling vaiable is based on physial insight (Astom & Wittenmak 989). The gain-sheduling design poess is fequently iteative, ith the ontolle evised in the light of subsequent analysis until a satisfatoy design is ahieved. The effetiveness of the gain-sheduled design appoah depends on the dynami haateistis of the nonlinea system, omposed of the nonlinea plant and the nonlinea gain-sheduled ontolle, being elated to those of the membes of an assoiated family of linea systems, omposed of the plant lineaisations and oesponding loal linea ontolles. The eisting esults elating the dynami haateistis of a nonlinea system to those of an assoiated family of linea systems is evieed in Leith & Leithead (997b,) and summaised belo. Seies epansion lineaisation theoy is ell established but is stitly onfined to the dynami analysis, loally to a single tajetoy o equilibium opeating point, of smooth nonlinea systems. When the family of equilibium opeating points an be paameteised by the input to the nonlinea system (as distint fom the sheduling vaiable), fozeninput tehniques ate fo the analysis of smooth nonlinea systems elative to a family of equilibium opeating points and elate the stability of a nonlinea system to the stability of a family of fozen-input nonlinea systems. A slo vaiation equiement is neessay hih seems to be inheent to this type of analysis, impliitly estiting the lass of alloable inputs and initial onditions; that is, impliitly estiting the tajetoies to emain suffiiently lose to the equilibium opeating points. In ode to elate the stability of the nonlinea system to the popeties of a family of linea time-invaiant systems, a futhe epliit estition on the alloable tajetoies is neessay to ensue they emain suffiiently lose to the equilibium opeating points that seies epansion lineaisations ae valid. This latte estition is not a pioi neessay yet may be vey stong sine the neighbouhoods ithin hih the seies epansion lineaisations ae valid may, in geneal, be eessively small. The utility of fozeninput theoy is, thus, somehat diminished sine it may imply a high degee of unneessay onsevativeness. Seies epansion lineaisation theoy and fozen-input theoy onside only the stability popeties of the nonlinea system and povide little diet insight into othe dynami popeties, suh as the tansient esponse. When the sheduling is not ontinuous, fe tehniques, othe than etensive simulation testing, appea to be available fo analysing the dynami behaviou of the ontolled system. Although fozen-input theoy an suppot the analysis of a nonlinea gain-sheduled ontol system, it povides little insight into the ontolle design poedue sine the fozen-input epesentation of the ontolled system is quite distint fom the mied seies-epansion/fozen-sheduling vaiable epesentation employed in step 3 of the design poedue. Seies epansion lineaisations of the plant ae employed but the oesponding loal ontolle designs ae fozen-sheduling vaiable lineaisations of the esulting nonlinea ontolle. In ontast, hen analysing the dynami behaviou of the ontolle loally to a single equilibium opeating point, the seies epansion lineaisation is employed instead of the fozen-sheduling vaiable lineaisation. Futhemoe, sine the sheduling vaiable, ρ, is vaying in the nonlinea ontolle but onstant in the loal designs, the nonlinea ontolle need not have the designed dynamis, loally to an equilibium opeating point. Moeove, the analysis of the ontolled system in the viinity of the family of equilibium opeating points does not edue to eithe the seies epansion analysis (see, fo eample Shamma 988 p) o the mied seies-epansion/fozen-sheduling vaiable analysis employed in the design poedue. The analysis of onventional gain-sheduling design by means of eisting esults, elating the dynami haateistis of a nonlinea system to those of an assoiated family of linea systems, is, theefoe, athe omple and ineffiient. The use of a vaiety of diffeent loal appoimations obsues insight and is suely unneessay. In addition, the analysis is onfined to stability popeties and does not dietly etend to othe dynami haateistis suh as the tansient esponse. Hene, the eisting theoy does not povide an adequate fameok to suppot the analysis and design of gainsheduled ontolles. 3. Veloity-based lineaisation families An altenative appoah, not disussed in setion, to the analysis of a nonlinea system by elating its dynami haateistis to those of an assoiated family of linea systems, is developed in Leith & Leithead (997b). Conside, the behaviou of the nonlinea system, (), hen thee ae no estitions on the lass of alloable inputs and initial onditions. The solutions to () may tae tajetoies anyhee in Φ and ae not onfined to the viinity of eithe a single equilibium opeating point o the

4 lous of equilibium opeating points. Suppose that the nonlinea system is evolving along a tajetoy, ((t), (t)), in Φ and at time, t, the tajetoy has eahed the point, (, ). It is emphasised that the point, (, ), need not be an equilibium opeating point and, indeed, may lie fa fom the lous of equilibium opeating points. Fom Taylo seies epansion theoy, the subsequent behaviou of the nonlinea system, (), an be appoimated, loally to (, ), by the fist ode epesentation, δ F(, ) F(, ) δ F(, ) δ (9) δ y G(, ) δ G(, ) δ () δ -, y y δ y, δ, δ () povided δ N δ N, hee the neighbouhoods, N and N, of, espetively, and ae suffiiently small. When (9)-() and () have the same initial onditions, (, ), the solution to (9)- () is, initially, tangential to the solution of () and, indeed, loally to time t, povides a fist-ode appoimation to (t) and a seond-ode appoimation to (t) (Leith & Leithead 997b,). The solution to the fist-ode seies epansion, (9)-(), povides a valid appoimation only hile the solution, (t), to the nonlinea system emains in the viinity the opeating point, (, ). Hoeve, the solution, (t), to the nonlinea system need not stay in the viinity of a single opeating point. Conside, theefoe, the appoimation to (t) ove a time inteval, [t,t ], obtained by patitioning the inteval into a numbe of shot sub-intevals. Ove eah sub-inteval, the appoimate solution is the solution to the fist-ode seies epansion elative to the opeating point eahed at the initial time fo the sub-inteval (ith the initial onditions hosen to ensue ontinuity of the appoimate solution). The numbe of loal solutions employed is dependent on the duation of the sub-intevals, but the loal solutions ae no auate to seond ode; that is, the appoimation eo is popotional to the duation of the sub-inteval ubed. Hene, as the numbe of sub-intevals ineases, the appoimation eo assoiated ith eah apidly deeases and the oveall appoimation eo edues. Indeed, the oveall appoimation eo tends to zeo as the numbe of sub-intevals beomes unbounded (Leith & Leithead 997b,). Hene, the family of fist-ode seies epansions, ith membes defined by (9)- (), an povide an auate appoimation to the solution of the nonlinea system. Moeove, this appoimation popety holds thoughout Φ and is not onfined to the viinity of a single equilibium opeating point o even of the lous of equilibium opeating points. The state, input and output tansfomations, (), depend on the opeating point elative to hih the seies epansion is aied out. Combining (9)and () ith the loal input, output and state tansfomations, (), eah membe, (9)-(), of the family of fist-ode epesentations may be efomulated as, {F(, ) - F(, ) - F(, ) } F(, ) F(, ) () y {G(, ) - G(, ) G(, ) } G(, ) G(, ) (3) The state, input and output is no the same at evey point in Φ, but the dynamis, ()-(3), ae nonlinea. By diffeentiating, ()-(3) may be efomulated in the equivalent veloity-based fom,, F(, ) F(, ), y G(, ) G(, ) (4) With appopiate initial onditions, namely, (t ) (t ), (t ) (t ) ( t ) F(, ), y (t ) y(t ) G(, ) (5) the tansfomed system is dynamially equivalent to the oiginal system. Hoeve, in ontast to ()-(3), the tansfomed system, (4), is linea. Thee eists a veloity-based lineaisation, (4), fo evey point in Φ. Hene, a veloity-based lineaisation family, ith membes defined by (4), an be assoiated ith the nonlinea system, (). The elationship beteen the nonlinea system and its veloity-based lineaisation family is diet. Diffeentiating (), an altenative epesentation of the nonlinea system is, F(, ) F(, ), y G(, ) G(, ) (6) Dynamially, (6), ith appopiate initial onditions oesponding to (5), and () ae equivalent Evidently, the veloity-based lineaisation, (4), is simply the fozen fom of (6) at the opeating point, (, ). (When F(, ), y G(, ) is invetible at evey opeating point, (, ), in an appopiate neighbouhood enlosing the lous of equilibium opeating points, so that may be epessed as a funtion of, and y, then the tansfomation elating (6) to () is, in fat, algebai). The solutions to the membes of the family of veloity-based lineaisations, (4), an be pieed togethe to appoimate the solution to the nonlinea system, (6). In this ase, the (t) ae still seond-ode appoimations to the (t) but the (t) ae fist-ode appoimations to the (t). Hoeve, it is staightfoad to sho that the piee-ise appoimation onveges to the eat solution (Leith & Leithead 997b,).

5 In ontast to the pevious appoahes disussed in setion, the veloity-based lineaisation analysis has seveal advantages. Thee eists a lineaisation of the nonlinea system at evey opeating point and not just the equilibium opeating points. Stability onditions ae deived fo nonlinea systems hih avoid the estitions, to tajetoies lying ithin an unneessaily, pehaps eessively, small neighbouhood about the lous of equilibium opeating points, inheent to pevious appoahes based on fozen-input theoy (Leith & Leithead 997b,). A estition on the alloable lass of inputs and initial onditions is still equied. Hoeve, in ontast to pevious esults, it is emphasised that this estition is puely a onsequene of the slo vaiation equiement and, in this sense, is a eak as possible. Indeed, fo systems hee thee is no estition on the ate of vaiation, the analysis is global in natue. Hene, the stability onditions deived using the veloity-based lineaisations ae inheently muh less onsevative than those obtained peviously. The stability analysis is also etended to inlude nonlinea systems ith non-smooth dynamis, suh as gain-sheduled ontolles hih sith beteen loal ontolles athe than employing smooth intepolation (Leith & Leithead 997b,). Futhemoe, the veloity-based lineaisation analysis is not onfined to stability. Sine the membes of the family of veloity-based lineaisations an be pieed togethe to appoimate the solution to a nonlinea system, the tansient behaviou of the nonlinea system an also be investigated. This appoimation is not onfined to the viinity of the equilibium opeating points but is valid thoughout the opeating envelope, inluding duing tansitions beteen equilibium opeating points and at opeating points hih ae fa fom equilibium. Consequently, the veloity-based lineaisation theoy has onsideable potential fo suppoting the design and analysis of gain-sheduled ontolles. 4. Gain-sheduled design using veloity-based lineaisation families The equiement is to dietly eploit the advantages of veloity-based lineaisations; patiulaly, to avoid the estition to opeation in the viinity of the equilibium opeating points hih is inheent in eisting gain-sheduling design appoahes. Sine the veloity-based lineaisation family assoiated ith a nonlinea plant desibes the dynami behaviou at evey opeating point, not just equilibium opeating points, it lealy has the potential to meet this equiement. Conside the nonlinea plant, (), and the nonlinea ontolle F (, ), y G (, ) (7) m hee R denotes the input to the ontolle, y R p the output and R n the state. Sine the equiement is to design a feedbak ontolle, it is assumed ithout loss of geneality that the input veto,, to the plant inludes the output, y, of the ontolle and the input veto,, to the ontolle inludes the output, y, of the plant. Let p denote the veto onsisting of the elements of hih ae not elements of y, and let denote the veto onsisting of the elements of hih ae not elements of y. In addition, it is assumed that the that the invese plant mapping fom F(,) to (,) is bounded; that is, is bounded hen F(,) and ae bounded. By diffeentiating, the plant may be efomulated in veloity-based fom as y y, A(ρ) B( ρ) B ( ρ), y C(ρ) D( ρ) D ( ρ) (8) p and the ontolle may be efomulated in veloity-based fom as y, A (ρ ) B ( ρ ) B ( ρ ), y C (ρ ) D ( ρ) D ( ρ) hee A( ρ) F(, ), B ( ρ) F(, ), B( ρ) F(, ) p y p C( ρ) G(, ), D ( ρ) G(, ), D( ρ) G(, ) p p y A ( ρ ) F (, ), B ( ρ ) F (, ), B ( ρ ) F (, ) y C ( ρ ) G (, ), D ( ρ ) G (, ), D ( ρ ) yg (, ) p p p y (9) and ρ(,), ρ (, ) embody the dependene of the dynamis on the states and inputs of the plant and ontolle, espetively. The ombined losed-loop dynamis ae depited in figue a. Assuming yg(,), y G (, ), ith elated to y and elated to y as desibed above, has a solution, y H (,,, ), the p veloity-based fom fo the losed-loop system may be epesented dietly in tems of the veloity- ()

6 based fom of the plant, (8), and the veloity-based fom of the ontolle, (9), as depited in figue b (see Appendi). The veloity-based lineaisation families assoiated ith the plant and the ontolle onsist simply of the fozen foms of, espetively, (8) and (9), obtained fo onstant values of ρ and ρ. In addition, eah membe of the veloity-based lineaisation family fo the losed-loop system may be obtained by enlosing the appopiate membes of the plant and ontolle families in a feedbak loop. Given this diet elationship beteen the veloity-based fom of the nonlinea systems and thei assoiated veloity-based lineaisation families and the stong oespondene in thei dynami behaviou as disussed in setion 3, the veloity-based lineaisation families onstitute a muh moe appopiate fameok fo the analysis and design of gain-sheduled ontolles than onventional appoahes. The foegoing analysis suggests the folloing gain-sheduling design poedue.. Detemine the veloity-based lineaisation family assoiated ith the nonlinea plant dynamis.. Based on the plant veloity-based lineaisation family, detemine the equied ontolle veloitybased lineaisation family suh that the esulting losed-loop family ahieves the pefomane equiements. Sine eah membe of the plant family is linea, onventional linea design methods an be utilised to design eah oesponding membe of the ontolle family. 3. Realise a nonlinea ontolle oesponding to the family of linea ontolles designed at step. The veloity-based fom of the ontolle an be obtained dietly fom the family of linea ontolles by simply pemitting the ρ to vay ith the opeating point. Sine the veloity-based fom of the system, omposed of the nonlinea plant, (), togethe ith the veloity-based fom of the ontolle, (9), is idential to the veloity-based fom of the system omposed of the veloitybased fom of the plant, (8), togethe ith the veloity-based fom of the ontolle, (9), see the Appendi, an altenative to the ealisation of figue a is that shon in figue ith the veloitybased fom of the ontolle. The latte has the advantage of avoiding the need to detemine a nonlinea ontolle, (7), oesponding to the veloity-based fom, (9). This design poedue etains a divide and onque philosophy and maintains the ontinuity ith linea design methods hih is an impotant featue of the onventional gain-sheduling appoah. Hoeve, in ontast to the onventional gain-sheduling appoah, the esulting nonlinea ontolle is valid thoughout the opeating envelope of the plant, not just in the viinity of the equilibium opeating points. This etension is a diet onsequene of employing the veloity-based lineaisation fameok athe than the onventional seies epansion lineaisation about an equilibium opeating point. With egad to step 3 of the design poedue, it should be noted that thee ae a numbe of issues hih must be onsideed hen detemining the nonlinea ontolle ealisation oesponding to the family of linea ontolles designed at step. In patiula, the output, y, of the ontolle is an input to the plant and the input,, of the ontolle is an output fom the plant. Hene, the value of and y at an equilibium opeating point of the plant is ( o, y o ) ith o dependent on y o via the plant. Hoeve, sine is the ontolle input and y is the ontolle output, ( o, y o ) must also be an equilibium opeating point of the ontolle ith y o dependent on o via the ontolle. Requiing onsisteny imposes, in geneal, a stong estition on the alloable nonlinea ontolles. Hoeve, this estition is iumvented by adopting the veloity-based ealisation of figue sine, in equilibium, the output of the diffeentiation tem befoe the ontolle and the input to the integal tem afte the ontolle ae both zeo. Of ouse, the pesene of a deivative and integal ation on the foad path in the veloity-based ealisation of figue equies to be teated ith some ae. Hoeve, hen the ontolle ontains integal ation, the diffeentiation opeato at the input and the pue integato ithin the ontolle may be fomally absobed togethe so that the input to the ontolle beomes athe than. The integation of the ontolle output then epliitly povides the equied integal ation (Leith & Leithead 997a). A futhe issue that must be addessed is the most appopiate manne in hih to implement the sheduling vaiable, ρ. It is staightfoad to implement ρ hen it is a funtion of y and alone. When ρ is also a funtion of, a numbe of appoahes an be adopted to obtain an appopiate ealisation of the sheduling vaiable. Fo eample, hen [F G ] T is invetible suh that may be epessed as a funtion of, and y, then so an ρ. It should be noted that, in these iumstanes, the diet fomulation, (7), is elated to the veloity-based fomulation, (9), by an algebai tansfomation. These issues ae disussed in detail in It is emphasised that this opeation is puely fomal in natue: no unstable pole-zeo anellation ous ithin the implemented ontolle.

7 Leith & Leithead (997a) and the implementation appoahes disussed thee, hilst developed in the ontet of the onventional gain-sheduling design appoah, may be eadily etended to the lass of ontolles onsideed hee. Eample Conside the fist ode plant ith dynamis G( ), y () hee G(s)tanh(s).s. The equiement is to design a ontolle suh that the losed-loop system has a ise time of aound.3 seonds ith less than 5% oveshoot in esponse to demanded step hanges in y of magnitude less than units. At an equilibium opeating point, ( o, o, y o ), G( o - o ) () hih equies that o - o (3) Hene, the seies epansion lineaisation of () elative to the equilibium opeating point, ( o, o, y o ), is δ G( ) δ G( ) δ, δy δ (4) δ o, δ o, y δy yo (5) Sine the fist deivative of the nonlinea funtion, G, is G( s). tanh s (6) the seies epansion lineaisation at an equilibium opeating point may be efomulated as δ. δ. δ, δy δ (7) δ o, δ o, y δy yo (8) Hene, based on the onventional seies epansion lineaisation at an equilibium opeating point, an appopiate loal ontolle is the PI-type ontolle δ 5 δ 5 δ δe, δ K K δ δ (9) δ δe e e, δ (3) o o K 5 ith ey ef -y, K o 3.86, K.. The tansfe funtion of the ontolle, (9), is K o s s 5. The Bode plot of the losed-loop tansfe funtion obtained by ombining (7) and (9) is depited in figue. The dynamis, (7), ae the same at evey equilibium opeating point and so the ontolle, (9), may be employed at evey equilibium opeating point. Oing to the integal ation in the ontolle,e o is zeo and o is impliitly geneated by the feedbak. Hene, on the basis of the family of lineaisations at the equilibium opeating points, a linea PI-type ontolle seems to be appopiate; namely, e, K K 5 5 (3) The step esponse of the losed loop system obtained by ombining the nonlinea plant, (), ith the linea ontolle, (3), is depited in figue 3. Evidently, the linea ontolle does not ahieve the equied pefomane. Indeed, simulation esults indiate that, fo step demands geate than appoimately.3 units, this ontolle is unable to satisfy the oveshoot equiements. In ode to inopoate infomation about the plant dynamis at non-equilibium opeating points into the ontolle design, efomulate the nonlinea plant, (), by diffeentiating, as G( ) G( ), y (3) The veloity-based lineaisation family assoiated ith the nonlinea plant, (), onsists of the fozen foms of (3) obtained hen and ae onstant, G( ) G( ), y (33) The equied veloity-based lineaisation family of the ontolle is detemined by using linea methods to design a loal ontolle fo eah of the membes of the plant veloity-based lineaisation family. Employing a PI-type ontolle stutue one again, onside the linea ontolle family 5 5 e, K K (34)

8 hee K o -.4./ G( - ) and K./ G ( - ). At equilibium opeating points, the membes of the linea ontolle family oespond to the ontolle dynamis, (3), detemined peviously. Hoeve, at non-equilibium opeating points, hee - is non-zeo, the gains K o and K ae no diffeent fom thei equilibium values and ae designed to ompensate fo the vaiation in the dynamis of the membes of the plant veloity-based lineaisation family. Sine the ontolle ontains integal ation, a nonlinea ontolle ith the veloity-based lineaisation family, (34), an be obtained by dietly implementing the veloity fom of the ontolle, see figue 4. The step esponse of the losed-loop system theeby obtained is depited in figue 5. It an be seen that the pefomane equiements ae met fo the full ange of step demands. 5. Plants satisfying the etended loal linea equivalene ondition Assume that the plant dynamis ae of the fom A B f(ρ), y C D g(ρ) (35) hee, R m, y R p, R n, ρ(, ) R q, A, B, C, D ae onstant maties, f( ) and g( ) ae diffeentiable nonlinea funtions and ρ, ρ ae funtions of ρ alone. In addition, assume that ρ minimally paameteises the lous of equilibium opeating points, ( o, o ), of the plant. The vaiable, ρ(,), equals the onstant value, say ρ, upon a sufae of o-dimension q in Φ and ρ and ρ ae onstant ove eah sufae. Hene, the nomal to eah sufae is idential at evey point on the sufae and eah sufae is, theefoe, affine. Moeove, to ensue that ρ is a unique funtion of and, these sufaes must be paallel fo all ρ. Consequently, it may in fat be assumed, ithout loss of geneality, that ρ and ρ ae onstant. The veloity-based lineaisation family assoiated ith the nonlinea plant, (35), is ( ( ) ) ( ( ) ) ρ ρ ρ y ( C g( ) ) ρ ρ ρ ρ ρ ( C g( ρ ) ρ) A f B f ρ ρ The membes of the family ae paameteised by ρ; that is, the veloity-based lineaisation is the same at opeating points lying on a sufae of onstant ρ. The union of the sufaes of onstant ρ oves the entie opeating spae, Φ, and, sine ρ minimally paameteises the lous of equilibium opeating points, eah sufae of onstant ρ intesets the equilibium lous at a unique point. Eah opeating point in Φ is, theefoe, assoiated, via a sufae of onstant ρ, ith an equilibium opeating point hih has the same veloity-based lineaisation. Hene, ρ may be intepeted as the sheduling vaiable assoiated ith the plant, in the sense that, at any opeating point in Φ, ρ indiates hih membe of the family of equilibium lineaisations of the plant is valid. Fo the lass of plants onsideed hee, the veloity-based lineaisation of the plant at the equilibium opeating points, hen taken togethe ith the sheduling vaiable, ρ, ompletely detemines the veloity-based lineaisation family assoiated ith the plant. The onventional seies epansion lineaisation of the nonlinea plant, (35), elative to the equilibium opeating point, ( o, o,y o ), at hih ρ equals ρ, is δ ( A f( ρ ) ρ) δ ( B f( ρ ) ρ) δ ρ ρ δy ( C g( ρ ) ρ) δ ( C g( ρ ) ρ) δ (37) ρ ρ δ o ( ρ ), δ o( ρ ), y δy y o( ρ ) It an be seen that, hilst the state, input and output diffe, the seies epansion lineaisation at an equilibium opeating point, (37), has the same fom as the oesponding veloity-based lineaisation, (36), at that equilibium opeating point. Hene, povided some ae is taken, it is possible to employ eithe type of lineaisation to haateise the plant dynamis. Reall that the onventional gainsheduling design appoah utilises the seies epansion lineaisations of the plant at the equilibium opeating points. Clealy, fo the lass of plants onsideed hee, the potential eists fo employing the onventional gain-sheduling design appoah to obtain a nonlinea ontolle hih is valid not only in the viinity of the equilibium opeating points but thoughout the opeating envelope. In that ontet, the pesent fameok povides igoous insight into the ole of the hoie of ontolle sheduling vaiable and ealisation: the natue of the veloity-based lineaisation family assoiated ith the esulting nonlinea ontolle and so the ontolle dynamis, patiulaly at non-equilibium (36)

9 opeating points, ae influened by them. Clealy, the ontolle dynamis need to be ompatible ith the oesponding plant dynamis at non-equilibium opeating points. The lass of plants onsideed hee onsists peisely of those satisfying the etended loal linea equivalene ondition oiginally poposed by Leith & Leithead (996,997a) in the ontet of detemining appopiate ealisations fo gain-sheduled ontolles. It is natual to selet a ontolle ith a nonlinea stutue hih eflets that of the plant and so it is attative to equie, fo the lass of plants onsideed hee, that the ontolle also satisfies the etended loal linea equivalene ondition. In these iumstanes, the gain-sheduling design poedue poposed in setion 4 speialises to:. Detemine the lineaisation of the plant at eah equilibium opeating point. In addition, detemine the sheduling vaiable, ρ, assoiated ith the plant; typially, this infomation might be deived fom physial undestanding of the plant.. Selet an appopiate sheduling vaiable fo the ontolle. An obvious hoie is to employ the plant sheduling vaiable, ρ, o an estimate theeof. Hoeve, an altenative hoie might be suggested by othe onsideations. 3. Design a suitable linea ontolle fo eah membe of the family of plant equilibium lineaisations. In ode to ensue the eistene of a oesponding ontolle satisfying the etended loal linea equivalene ondition, the vaiations beteen the membes of the esulting family of linea ontolles should be ompatible ith the hoie of sheduling vaiable. The ompatibility equiement is not ovely estitive in geneal, see Leith & Leithead (997a). 4. Realise a nonlinea ontolle, ompatible ith the hoie of sheduling vaiable and family of linea ontolles detemined at steps and 3, hih satisfies the etended loal linea equivalene ondition (see Leith & Leithead 996, 997a). Eample Conside the nonlinea plant, depited in figue 6, ith seond-ode dynamis a a ( ) b b A B( ), y (38) hee A, B ae diffeentiable nonlinea funtions ith sa(s)>, B(s)> s R. The nonlinea plant, (38), is of the fom, (35), ith ρ equal to, and satisfies the etended loal linea equivalene ondition. The veloity-based lineaisation of the nonlinea plant at the geneal opeating point, (,, ), is, a b A( ) a, B( ) b y (39) At an equilibium opeating point, (,, ),, b a A B a A ( ) ( ), ( ) (4) and so the equilibium opeating points may be paameteised by. The veloity-based lineaisation at the equilibium opeating point at hih equals is, a b A( ) a, B( ) b y (4) Hene, the veloity-based lineaisation at the geneal opeating point, (,, ), oesponds peisely to the veloity-based lineaisation at the equilibium opeating point at hih equals. The seies epansion lineaisation at the equilibium opeating point at hih equals is δ a δ A( ) a δ δ, δ δ δ b δ B( ) b y (4) δ δ, δ, δ, y δy y (43) Whilst the state, input and output of (4) and (4) diffe, it is lea that (4) has the same fom as (4). Employing the onventional gain-sheduling design appoah, a linea ontolle is designed fo eah membe of the family of seies epansion lineaisations, (4), at the equilibium opeating points. Conside the family of linea ontolles

10 δ δ δ 3 A( ) δ B( ) δ δe, δ δ, δ 3 δ δ 3 δe δy δy (44) The losed-loop tansfe funtion obtained by ombining (4) and (44) is (45) 3 s bs as The plant input,, is a natual hoie of ontolle sheduling vaiable sine it is also the plant sheduling vaiable. Employing one of the appoahes poposed in Leith & Leithead (996, 997a), an appopiate nonlinea ontolle ealisation is 3 A( ) B( ) e,, 3 3 ef e y y (46) It may be shon that the nonlinea ontolle, (46), satisfies the etended loal linea equivalene ondition ith the sheduling vaiable,, and the family of seies epansion lineaisations elative to the equilibium opeating points, (44) (Leith & Leithead 996, 997a). The fom of the ontolle ealisation is lealy dietly elated to that of the linea family, (44). In Leith & Leithead (996, 997a), ontolle ealisations of the fom, (46), (and moe geneal foms of ealisation) ae deived to minimise the slo vaiation onditions inheent to the onventional gain-sheduling appoah. Hoeve, in the moe geneal fameok onsideed hee, it is evident that (46) is simply a thid-ode eample, ith the nonlineaity puely a funtion of, of the veloity-based ontolle, (9). Hene, the veloity-based lineaisation family of the ontolle onsists of the fozen foms of (46) and the veloity-based lineaisation family assoiated ith the losed-loop system is obtained by ombining (39) and (46) (o, altenatively, (4) and (46)). The nonlinea ontolle, (46), although designed by the onventional gain-sheduling appoah of using only the plant equilibium lineaisations, is valid thoughout the opeating envelope, not just in the viinity of the equilibium opeating points. Indeed, it is staightfoad to sho that the ombined plant and ontolle dynamis onsist of linea dynamis ith tansfe funtion, (45), and an eponentially stable, unobsevable nonlinea omponent. ef 6. Conlusions In this pape, the veloity-based lineaisation fameok is employed to develop a novel igoous appoah to gain-sheduling design. The appoah addesses many of the defiienies of the onventional gain-sheduling design appoah. Whilst etaining ontinuity ith linea methods, hih is ental to the onventional gain-sheduling methodology, the appoah Enables knoledge onening the plant dynamis at non-equilibium opeating points to be inopoated dietly and igoously into the ontolle design. In ontast, the onventional gainsheduling appoah utilises only equilibium lineaisations of the plant and so is inheently estited to situations hee only athe small and/o sloly-vaying ontol demands and distubanes ae enounteed. Enables tansient pefomane equiements, instead of stability alone, to be onsideed dietly duing the ontolle design. Enompasses both smooth and non-smooth sheduling ithin the same analysis and design fameok. Employs a steamlined analysis and design fameok hih utilises a single type of lineaisation thoughout. Consequently, in ompaison ith the onventional gain-sheduling appoah, the design poedue poposed hee is both staightfoad and oneptually appealing. These benefits stem dietly fom adoption of the veloity-based lineaisation fameok fo the analysis of nonlinea systems. Aknoledgement D.J.Leith gatefully aknoledges the geneous suppot povided by the Royal Soiety fo the ok pesented. Refeenes

11 ASTROM, K.J., WITTENMARK, B., 989, Adaptive Contol. (Addison-Wesley). HYDE, R.A., GLOVER, K., 99, A Compaison of Diffeent Sheduling Tehniques fo H Contolles. Poeedings of the Institute of Measuement & Contol Symposium on Robust Contol, Cambidge. HYDE, R.A., GLOVER, K., 993, The Appliation of Sheduled H Contolles to a VSTOL Aiaft. IEEE Tansations on Automati Contol, 38, -39. LEITH, D.J., LEITHEAD, W.E., 996, Appopiate Realisation of Gain Sheduled Contolles ith Appliation to Wind Tubine Regulation. Intenational Jounal of Contol, 65, LEITH, D.J., LEITHEAD, W.E., 997a, Appopiate Realisation of MIMO Gain Sheduled Contolles. Intenational Jounal of Contol, to appea. LEITH, D.J., LEITHEAD, W.E., 997b, Dynami Analysis of Gain-Sheduled & Nonlinea Systems Using Veloity-Based Lineaisation Families. Poeedings of Wokshop on Multiple Model Methods in Modelling & Contol, Tondheim, Noay. LEITH, D.J., LEITHEAD, W.E., 997, Gain-Sheduled & Nonlinea Systems: Dynami Analysis by Veloity-Based Lineaisation Families. Intenational Jounal of Contol, submitted. RUGH, W.J., 99, Analytial Fameok fo Gain-Sheduling. IEEE Contol Systems Magazine,, SHAMMA, J.S., ATHANS, M., 99, Analysis of Gain Sheduled Contol fo Nonlinea Plants. IEEE Tansations on Automati Contol, 35, Appendi Conside the nonlinea system ith inputs, and z, F(,, z), y G(,, z) (47) Tansfoming into veloity-based fom, (47) is equivalent to F(,,z) z F(,,z) z F(,,z) (48) y G(,,z) z G(,,z) z G(,,z) Assuming that yg(,,y) (49) has a suitable solution yn(,) (5) the system, (47), is enlosed is a feedbak loop by setting zy. The esulting losed-loop system is M(, ), y N(, ) (5) ith M(, ) F(,, N(, )) (5) Tansfoming into veloity-based fom, (5) is equivalent to M(,) M(,) (53) y N(,) N(,) Combining (49) and (5) N(, ) G(,, N(, )) (54) Hene, M(, ) F(,, N(, )) F(,, N(, )) N(, ) z M(, ) F(,, N(, )) F(,, N(, )) N(, ) z N(, ) G(,, N(, )) G(,, N(, )) N(, ) z N(, ) G(,, N(, )) zg(,, N(, )) N(, ) and, by substituting (55) into(53), the losed-loop system, (53), an be dietly efomulated as F(,,z) z F(,,z) z F(,,z) (56) y G(,,z) z G(,,z) z G(,,z) z y N(,) Sine N(,) satisfies (54), it is lea that (56) is the system obtained hen the system, (48), is enlosed in a feedbak loop by setting zy. It follos that the veloity-based fom of a losed-loop system is idential to the system obtained by enlosing the veloity-based fom of the open-loop system in a feedbak loop. (55)

12 Conside, no, the nonlinea system F (,, z ), y G(,, z ) (57) fo hih the veloity-based fom is F (,,z ) z F (,,z ) z F (,,z ) (58) y G (,,z ) z G (,,z ) z G (,,z ) and the nonlinea system F (,, z ), y G (,, z ) (59) fo hih the veloity-based fom is F (,,z ) z F (,,z ) z F (,,z ) (6) y G (,,z ) z G (,,z ) z G (,,z ) The systems, (57) and (59), ae asaded togethe by setting z y. The esulting system is F(,, z), y G(,, z) (6) hee,, z z, y y (6) F (,, z ) F(,, z) G z G G z F (,, G (,, z )), (,, ) (,, (,, )) Tansfoming into veloity-based fom, (6) is equivalent to F (,, z ) F (,, z ) G (,, z ) F (,, z ) z z z F (,, z ) z F (,, z ) G (,, z ) z F (,, z ) F (,, z ) G (,, z ) F (,, z ) z y y z G (,, z ) G(,, z ) G (,, z ) G (,, z ) G (,, z ) z z z G (,, z ) G (,, z ) G (,, z ) z z y G(,, z ) Evidently, (63) is just the system obtained hen the systems, (58) and (6), ae asaded togethe. It follos that the veloity-based fom of a system onsisting of to asaded sub-systems is idential to the system obtained by asading togethe the veloity-based foms of the to sub-systems. Futhemoe, eept fo the output being y athe than y, both ae equivalent to veloity-based system, (58), asaded togethe ith the nonlinea system, (59). (63)

13 p Contolle (Diet fomulation) F (, ) y G ( ) Plant (Diet fomulation) F(, ) y G(, ) (a) p d/dt Contolle (Veloity-based fomulation)!" #! # y " # A (ρ ) B ( ρ ) B ( ρ ) y C (ρ ) d/dt Plant (Veloity-based fomulation) A(ρ) B( ρ) B ( ρ) y C(ρ) D( ρ) D ( ρ) p p y y p p (b) p d/dt d/dt Contolle (Veloity-based fomulation) A (ρ ) B y C (ρ ) ( ρ ) B ( ρ ) y Plant (Diet fomulation) F(, ) y G(, ) () Figue Diffeent fomulations of nonlinea feedbak system. Note, the diffeentiation opeatos ae puely fomal in natue.

14 5 Gain db -5 Phase deg Figue Bode plot of open-loop tansfe funtion at the equilibium opeating points (Eample ). output, y Fequeny (ad/se) - 3 Fequeny (ad/se) time(se) Figue 3 Responses to steps of magnitudes.,.5 and. ith linea ontolle designed using equilibium infomation only (Eample ).

15 K o e - 5 K Figue 4 Nonlinea ontolle ealisation (Eample ) output,y time (se) 6 8 Figue 5 Responses to steps of magnitudes, 5 and ith nonlinea ontolle designed using off-equilibium infomation povided by plant veloity-based lineaisation family (Eample ).

16 B() A() - - y b a Figue 6 Nonlinea plant onsideed in Eample