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Low-Order Robust Model Matching Controller Design for SISO Plants
Copyri ghl © IF AC Illh T rien nial World Congress. T allinn . ES lo ni a. L' SS R. 1990
LOW-ORDER ROBUST MODEL MATCHING CONTROLLER DESIGN FOR SI SO PLANTS Yi-Sheng Zhong*, T. Eisaka** and R. Tagawa** *Dept. of Autolllatioll , T ;illgil lla Ullil 'enit)' . Beijillg. PR C ** Dept. of Elect rica l E llgill eerillg, H okkaido 1.; llil'e 1'5 it)" Sapporo , j apall
Two design lethods of low-order robust lodel latching controllers for SISO plants with paraleter and order variations are presented . It is assuled that the relative degree and the sign of the gain constant of the transfer function of the plant are fixed, the zeros of the plant are aSYIPtotically stable and the coefficients and the degrees of the denolinator and nUlerator polynOlials of the plant transfer function vary in step-wise fashion in bounded and known ranges. A low-order robust lodel latching controller consists of an ordinary lodel latching controller and a robust cOlpensator. The lodel latching controller is designed by the usual lethod for a reduced-order "reference plant" constructed suitably. The nOlinal and actual plants are considered as those the reference plant changed into. The robust cOlpensator is designed to reduce the influence of the plant characteristic variations frol the reference plant. It is shown that, by suitably designing the robust cOIPensator, one can ensure that the closed-loop systel with a low-order robust lodel latching controller has robust stability, robust steady-state and robust transient properties silultaneously. For given reference lodel and reference input, the order of a controller designed by these lethods is deterlined by the relative degree of the plant, in spite of the order of the plant.
~.
~.
Control systel design; linear systel; robust control; stability; tracking systels. fixed, where the relative degree is defined as the difference of the degrees of the denolinator and the nUlerator polynolials of the plant transfer function. The paraleters and the order of the plant, subject to the incan vary in variant properties (i), (ii) and (iii), step-wise fashion in any but bounded and known ranges. Two design lethods will be presented frol different starting-points. For given reference lodel and reference input, the order of a controller designed by these lethods is deterlined by the relative degree of the plant, independent of the plant order. It is shown that, the robust properties, i.e., robust stability, robust steady-state property and robust transient property of the closed-loop systel can be guaranteed silultaneously. This paper is organized as follows. In the next chapter, the design problel is fOrlulated, and assulptions on the plant and on the reference lode I are lade. Two design lethods are presented in Chapter 3. In Chapter 4, the robustness of the closed-loop systel is proved . Design procedures are sUllarized in Chapter 5, and an exalPle and silulation results are shown in Chapter 6.
1. INTRODUCTION
Much attention has been paid to the design problels of robust control SystelS, and various design procedures have been proposed under different conditions for different types of systels . We focus our attention on linear tile-invariant systels. The classical frequency-dOlain lethods are powerful tools for designing robust controllers for single-input single-output systels [lJ, and are extended to analyze and synthesize lulti-input lultioutput Systels [2,3,4,5J. Control systels designed by LQG techniques have at least ±60· phase nrgin, infinite increasing gain largin and 50 percent gain reduction tolerance [6, 7J. Such robustness will be lost when state observers are used in the ilplelentation [8J. Loop recovery techniques can be used to ilProve the robustness of LQG control systels with observers [9,10,l1J. Unfortunately, LQG control systels have poor robustness to plant paraleter variations [12J . The HOO-optilal theory, proposed by G. Zales, is suit to deal with the design problels of robust control Systels for plants with uncertainties and/or disturbances belonging to prescribed set [13-17J. However, these lethods generally lead to high order controllers for high order plants. Controllers designed by LQG or HOO-optilal techniques have orders roughly the sale as the orders of the plants [6, 18-20J. There are several lethods to obtain low-order controllers. One can solve the linilal design problel to get low-order controllers [21J. One can use singular perturbation techniques [22J and paraleter optilization lethods [23J or apply sOle invariant properties [24J to derive low-order controllers. Model reduction techniques [25-29J can also be used to get low-order controllers. In our opinion, however, the existing results on the control systel designs for linilul phase plants are restrictive both in the robustness of the closed-loop properties and in the order reduction of the controllers. In this paper, we deal with the design problels of low-order robust lodel latching controllers for singleinput single-output linear tile-invariant plants. It is assuled that the plant to be controlled has the invariant properties: (i) its gain constant has a fixed sign, (ii) it is of linilul phase , and (iii) its relative degree is
tMill.Jm..
{oJ:
{I,2, ... ,n}
R: The set of all real nUlbers c: The set of all cOlplex nUlbers C o: {x : x e C, Re(x)O 2. PROBLEM STATEMENT
The single-input single-output (SISO) plant in nOlinal case is assUled to be described by the following equa tion. 191
Y=Gpo(s)U, Gpo(s)=kpoNpo(s)/Opo(s), (1) where Y and U are the output and the input of the plant respectively, kpo is a nonzero constant,Npo(s) E Pl [S,lpoJ and 0po(s) E P l[s,npo ). that is, Npo(s) and 0po(s) are Ionic polynOlials of degree lpo and degree npo respec t i ve ly . On the nOlinal plant (l), the following assuIPtions are aade. AsSUlption 1. (All) The degree npo and the relative degree nso := Rpo -lpO are known . The gain constant kpo and the coefficients of the polynOlials Npo(s) and Dpo(s) are known. The gain constant kpo is positive. (A12) Npo(s) is a Hurwitz polynolial, i.e, all the zeros of Npo(s) are in C_. (A13) Npo(s) and Dpo(s) are coprile. Obviously, the assUlption that kpo is positive causes no loss of generality . When kpo is negative, we can change the sign of the input. The reference lodel which specifies the desired control properties of the closed-loop systel is given by (2)
where YI is the output of the reference lodel and R is the reference input (external input) which is piecewise continuous, NI(s)E Pl[s, .. J and DI(s)E Ptls,IltJ. The reference lodel (2) is assUled to satIsfy the following assulptions . AsSUlption 2. (A2l) DI(s) is a Hurwitz polynolial . (3) (A22) p :=(nl-II)-nso~O. Under AssuIPtion 1 and 2, one can apply the usual lethods [30J to design a controller for the nOlinal plant such that the transfer function frOl R to Y of the closed-loop systel is identical with that of the reference lodel, i.e., WI(s), after having canceled sOle COlIon stable factors of its nUlerator and denOlinator polynolia!s. However, if a controller such designed is applied to the "actual" plant with dynaaic characteristics different frol the nOlinal plant, the control properties of the closed-loop systea lay becOle quite different frOl those specified by the reference lodel, and the closed-loop systea lay even lose stability. To deal with these problels, we lake sOle assuIPtions on the actual plant. Suppose the actual plant with paraleters and order having changed in step-wise fashion frol their nOlinal values is represented by
Y=Gp(s)U, Gp(s)=kpNp(s )/Dp(s), (4b) where Np(s) E P l[S,lpJ and Dp(s) E P ds,np]. The goal of this paper is to present two design lethods of low-order robust controllers for plants satisfying AssuIPtion 1 and 3 and reference lodels satisfYing AssuIPti on 2. 3. CONTROLLER DESIGN
We shall present two different design lethods. Scheme 1
First, we show a design lethod which applies the prior inforaation on the nOlinal plant. Let 1, if n~ is an odd posi ti ve integer and ,s 0:= [ 0po~s) has no real roots, (5) 0, else. It is easy to prove the following fact.
Leila 1. The transfer function Gpo(s) can always be separa ted as (6) Gpo(s)=Go(s)+Nro(s)/Gpo(s), where (i) Go(s) :=kpoNo(s)/Oo(s), (7a) (iD Dpo(s)=Do(s)Dl(S), DO(S)E Pl[s,n s+<5'0). (7b) (iii) a [Opo(s)J-a [Nro(s)J >ns ; (8) (iv) -1, if <5' 0=0, No(s):= \ _ s+c, If <5' 0=1, c is a positive constant such that Do(s) and No(s) are coprile. Reurk 1. (i) The constant c can allost be anY positive constant. In fact, it is only required that the c do not belong to the set {a: ID (-a)=O,aE R}. (ii) When ns is an od3 integer and 0po(s) has no real roots, to ensure all the coefficients of Do(s) being real,we have to set a [D o(s)J=n s +<5' o.To set a [N o(s)J=<5' 0 is to guarantee that the relative degree of No(s)/Do(s) is equal to that of Gpo(s). For the convenience of statelent, in the sequel, we call the systel with the transfer function Go(s) reference plant. Suppose the reference input is of the fora
Y=Gp(s)U, Gp(s)=N(s)/D(s), (4a) which satisfies the follOWing assuIPtions. AsSUIPtion 3. (A3l) N(s) E P [S,lpJ and D(s) E P [s,npJ, Ip:iiinl!; Ip:iiilo,np :iiino,where 10 and no are known and bounded positive integers. (A32) The coefficients of N(s) and Des) are bounded and vary in step-wise fashion in the known ranges S N and So respectively . (A33) N(s) 15 a Hurwitz polynOlial for all its coefficients in S N. (A34) There exists a fixed positive integer ns (called relative degree) , for all the coefficients of N(s) and D(s) in SN and SD respectively, such that lils-+ooGp(s)sns =:kp, w~ere kp (called gain constant) is a nonzero positIve constant taking value in the known and bounded range S k' and ns=npo-lpo=np-Ip. The assulptions (A3l) and (A34) ilply that the grees of N(s) and D(s) can vary, but their bounds known and their relative degree is fixed. The ranges and So are only requ ired bounded and knOllll. So, paraleter variations can be very large.
The equation (4a) can be rewritten as
deare SN the
192
R=Lr(s)/Or(s), where Dr(s)EP1{s,n J and a [Lr(s)J
(9)
U=[Ou(s)J-l[Nu(s)U+Ny(s)Y+krNr(s)R+VJ, (10) where V is an auxiliary input which will be given later; kr=k,/kpQ; Du(s) is a Ionic Hurwitz polynolial; Nu(s), Ny(s) and Nr(s) are polynOlials deterained by the fOllowing equations. Nr(s)=H(s)N~(s), (11) Do(s)Nu(s)+kpoNo(s)Ny(s)=Do(s)Du(s)-DI(s)H(s)No(s) ,(12) where H(s) is a Hurwitz Ionic polynolial of appropriate degree such that the follOWing degree relationships are satisfied.
Degree Relationship a [Dues) J=nl-n s +a [H(s) J; (13a) a [Du(s)J=lax{O,ns+,s o-l,na-ns} ; (l3b) a [Nu(s) J :iii a [Ou(s) J-l; (l3c) a [Ny(s)J:iii l" a [Du(s)J-l, ns=O, (13d) a [Du(s»). ns>O. The above degree relationships can always be satisfied by suitably choosing the degree of H(s). One can see that, if we set the auxiliary input V=O, then the control law given by eqn .(lO) is just an exact lodel latching control law for the reference plant. Since No(s) and Do(s) are coprile, the equation (12) has a solution for
Nu(s) and Ny(s). The actual plant can be represented in teras of the paraleters of the reference plant as
plant Go(s) frOl the nOlinal plant; any rational function can be chosen as our starting-point of design on a superplane in the rational function space specified by the invariant properties (i), (ii) and (iii). That is, we can use any rational function satisfying the Invariant Property as the "reference plant". Thus, we can choose
(14) °o(s)Y=kpoNQ(s)U+q, (15) q=[Oo(s)-Op{s)]Y-[kpoNo(s)-kpNp(s)]U. SUbstituting the U given by eqn.(IO) into (14), and using (11) and (12), one has
(26)
Y=WI(s)R+[OI(s)H(s)No(s)]-I{kpoNo(s)V+[Du(s)-Nu(s)]q}. (16) Here, we can consider the actual plant (4) to be one the reference plant changed into after dynaaics (paraaeters and degrees of its transfer function) having varied. The variable q given by eqn.(I5), called equivalent djstur~, represents the influence of the dynalical variations on the control properties of the closed-loop systel. We want to design the auxiliary input V to elilinate or reduce the influence of the equivalent disturbance q. If we set the auxiliary input V as
where H.(s) is any Ionic Hurwitz polynOlial of degree nl -n s ' and o.(s) is the denOlinator of the transfer function of the reference lodel (see eqn. (21». Obviously, the rational function "-(S)/OI(S) satisfies the invariant properties (i),(ii) and (iii). The description of the actual plant in terls of the polynolials "-Cs) and 0l(s) then becOles 01(s)Y=HI(s)U+q, (27) q=[OI(s)-Op(s)]Y-[HI(s)-kpNp(s)]U. (28) The control input U can be constructed as
V=-[kpoNo(s)]-I[Ou(s)-Nu(s)]q, (17) then, exact lodel latching is achieved in spite of the dynalic variations within the plant. However, the control law given by eqs.(IO) and (17) cannot be ilPlelented. So, a filter F(s), called robust filter, is introduced, and the auxiliary input V is constructed as (18) V=-F(s)Z, (19) Z=[kpoNo(s)]-I[Ou(S)-Nu(s)]q. The robust filter F(s) is of the fori:
(29) (30) where F(s) is a robust filter given by equations (20-24).
U=[H.(s)]-I[k,NI(S)R-F(s)q],
q=OI~s)Y-HI(s)U,
Since the polynolial H (s) can be chosen arbitrarily in the set of Ionic Hurwi'z polynOlials of degree nl-ns' the paraaeters to be deterlined of the controller described by equations (29) and (30) are only the fi (iE (~}). For given reference lodel WI(s) and reference input R, the structure of the controller is deterlined only by the relative degree ns ' which decides the degree of the polynolial HI(s). The choice of the paraaeters fi (iE (~) depends on the ranges of the dynalical (i.e., paraleter and degree) variations within the plant. The Order of the Controller Now, we cOlpare the orders, denoted by n, of the controllers designed by the lethods presented a~ove. For Schele I, ns=O, nr+nl' (31) nc= [ nr -1+.ax{2ns +o o-1.n I L ns>O; and for Schele 2, ns=O; nr+nl' (32) nc= { nr +n l -1. ns>O. For exaaple, in the case where ns=I, R=ro/s, roE R, WI(s)= al/(s+a l ), a. ER and al>O, 1+ 0 0 , for Schele I, n- { c- 1. for Schele 2, in spite of the order of the plant!
F(s)=Nf(s)/Of(s), (20) (21) °f(S)=(s+fI)nr-I(s+f2)(s+f3) , , t =lax{O,ns-I}, (22) (23) f3 »f2 »fI >0; (24a) Nf(S)=Of(s)-Or(s)Hr(s), Nf(s)E P[s.nr-I]. Hr(S)E PI[s, t]. (24b) where Ores) is the denOlinator of the Laplace transforl of the reference input (see eqn.(9». The sYlbol M»~" in eqn. (23) leans "Iuch greater" . FrOl equations (19), (15), (4) and (12), one can have Z=[OI(s)H(s)/ kpo+Ny(s)]Y-[Ou(s)-Nu(s)]U. (25) We call the part of the controller described by eqs. (18), (20) and (25) robust cQlpensator. Relark 2. (i) It is easy to see that the whole controller represented by equations (10). (18), (20) and (25) is realizable. (ii) The order of this controller is deterlined by the degrees of 01(S) and 0r(s), and the relative degree ns of the transfer function of the plant. and is independent of the degree of 0po(s) or 0p(s), i.e., the order of the nOlinal plant or the actual plant.
Reeark 3.
(i) We can see that the order of a controller designed by lethods presented above can even be lower than that of an exact lodel latching controller designed by the usual lethods [30] in the following cases:
Scheme 2
(a) ns=O and np>nr+nl+I; (b) ns>O and np>n r +lax{2n s-o o-I,n l } for Schele I, np>nr+nl for Schele 2.
The lain idea of the design lethod stated above is that, a lodel latching controller is first designed for the reference plant Go(s) derived frOl the nOlinal plant, the influence of the "dynalical variations" of the plant frOl the reference plant is represented by the equivalent disturbance q, then a robust cOlpensator is designed to reduce the influence of the equivalent disturbance q. The starting-point of this design lethod is the reference plant . The reference plant has the following properties. which are the sale as those of the plant. Invariant Property (i) The gain constant is positive. (i i) All the zeros are in C _. (iii ) The relative degree is equal to ns' FrOl the analysis on the properties of the closedloop Systels with a controller designed by the lethod stated above, we see that the locations of the zeros and poles of the reference plant is not ilportant, but the invariant properties (i), (ii) and (iii) are crucial. So, it has no need of deriving the reduced-order reference
(ii) Choosing a lower order reference lodel can lead to a lower order controller. However. since n.~ns' the order of a controller designed by the lethods presented above is finally deterlined by the relative degree ns of the plant for a given reference input.
4. ROBUSTNESS
In this chapter. we shall investigate the robust control properties, i.e., robust stability, robust steadystate property and robust transient property, of the closed-loop systel conSisting of the plant (4) and a controller designed by the lethods presented in the last chapter. 4.1 Robust Stability
A lella is first introduced. 193
r.e.a
2.
Let
v(s):=Of(S)+(kplkpo -I)Nf(S)-(s+fl)nr-l(s+f2 kplkpo)(S+t3) , =(l-kplkpo)[f2(S+fl)nr-l(stf3)' -Nf(s»). (41)
h(s)=sn+hn_ISn-I+ ... +hIS+ho. lIhere hi_I's (i£{n}) are real constants. Then. aax[ I a I : h(a)=O.
then. the equation (36). hence (33) has nr-I roots inside r(-fl. ~ Ifl). one root inside r(-Af2.Bt2) and , roots inside r(-f 3 • ~3f3)' Since r M• r(-fl. ~ Ifl)' r(-Af2. Bf2) and r(-t3' ~3f3) are pairllise disjoint. if conditions Cl. C2 and C3 are satisfied. then the equation (33) has ngtnf roots inSIde r M and r Obviously. all the roots are in C_. Hence. the proo of this theorel is done if lie can show that for any possible dynalical variations lIithin the plant. if the fi (i £ {J.}) are sufficiently large and fi+1 IS luch greater than fi (i £ {Z). conditions Cl. C2 and C3 can be satisfied. In the sequel. lie use the expression" a - r [,8)" to denote the fact that lil,B-+coa I ,8=r .lIhere a >0. ,8>0. r ;;: O. The condition (23) is thus expressed as
C }hl+l. one has I h(s)/sn 1;;:[ I sn I-I h _ISn-II- ... -1 ho I}I I sn I ;;: I-h l [ I s-I f+ ... +I s-n I} co . > I-hi [~ I S-1 I };;: o. i= I Hence. for I s I >~+l. I h(s) I ?f0. o a£
f'
The case of Schelle 1 FrOl (4). (10-12). (17). (18) and (25). it follows that the characteristic equation of the closed-loop systel designed by Schele I is the following. ~(s):=(kplkpo)Nf(S)Ng(s)+[Of(S)-Nf(S)}Og(s)=O.
(33)
Ng(s):=Np(s)OI(s)H(s). Og( s) : =Op(s) [Ou(s)-Nu(s) }-kpNp(s)Ny (s).
(42)
Let
IIhere (34) (35)
(43) (44)
Nf(s)=anr_ISnr-I+ ... +als+aO' Hr(s)=s' +b, -Is' -1+ ... +bls+b o '
The equation (33) can be rellritten as
FrOl (21) and (24). it can be shown that [Of(s)+(kp/kpo-I)Nf(s)}Og(s)+[Ng(s)-Og(s)}(kp/kpo)Nf(s)=0. (36) FrOl eqn.(13). one has and
11 [N g (s)}=11 [Og(s)} (=:ng ).
(37)
where ICn:=n! / [(n-I)!I!}.
Ng(s)£ P}ls.ngl.
(38)
(i) The Proof of Condition Cl FrOl the definition of f M• one can see that there exists a nonzero positive constant '1J such that for s £ rM. I Ng(s) I ;;: '1J ;>! O.
0g(s) £ PI[s.ng}'
Let nf:=11 [Of(s)}=nrt,. Then. the equation (33) has nf+ ng roots. Since Ng(s) is a HUrllitz polynOlial and the range of the coeffIcients of Np(s) is known. by Lella 2. one can deterline a positive constant ~H such that all the zeros of Ng(s) locate inside the half-circle rH given by rM:={s:
I s I :i ~J1. Re(s)=O. or I s I = ~ H. Re(s) :i0.
SE
Hence. frol (42). (45) and (46). one has that. for s£ f M• I (kp/kpo)Nf(S)Ng(s) I ;;: '1J (kp/k po ) I N~(s~ I - '1J (kp/kpo)[fl r- f2f3' J. and I [Of(S)-Nf(S)}Og(s) I -O[fl nr - I f 2f3' ).
(39)
c}.
By the prior inforaation on S k. one can also cons tants a. b £ R. a>b>O such that
deterline
here lie have used the fact that the coefficients and degrees of 0p(s). Np(s) and 0E(s) are bounded. Therefore. Condition I can be satisfied if f i (i £ {~}) are sufficiently large. and fit I is luch greater than fi (i £ {Z).
(40) Let ~ I and ~ 3 be constants in the interval (0.1). i. e .• O<~I
(ii) The Proof of Condition C2 FrOl the boundedness of the paraleters of the plant and the eqn.(37). for SE r(-fl. ~ Ifl). i.e .• s=zlfl. zl=-l+~lexp(je) (O:;oe<21t). one has
Theorel 1. For the plant described in nOlinal case by (I) and in actual case by (4). and the reference lodel given by (2). subject to AssuIPtion I. 3 and 2 respectively. if the controller represented by (10). (17). (18) and (25) is applied. and if the paraleters fi (i £ eU) are sufficiently large and f itl is luch greater than fi (i £ {Z). then. the closed-loop systel is robustly stable. and its poles locate inside r M• r(-fl' ~ If l). r(-Af 2. Bf2) and r(-f3. ~3f3) '
I 0f(S)+(kp/ kpo)Nf(S) }Og(s) I - I zll na lnr-l(kplk 0) [flnrtng-If2f3 ' }
;;:(1-~ l)ng~ lnr-I[fln~tng-
I [Ng(s)-Og(s)}Nf(S)kp/ kpo I -0 [flnrtng-lf2f3 ' }. Sililarb. for
I (kplkpo)Nf(S)Ng(s) I > I [Of(s)-Nf(S)]Og(s) I
.SE
and for
f(-A!2.Bf2). one has
SE r(-f3.~3f3).
I 0f(s)+(kp/kpo)Nf(s) }Og(s) I -
'1J
2 [f3ngt nrt ,
I [Ng(s)-Og(s)}Nf(S)kplk po I -O[f3 ng+nr +, J. the
I.
where '1J 1 and '1J 2 are nonzero positive constants independent of f i (i d~}). So. if fi (i £ {~}) are sufficiently large and fitl is luch greater than f i (i dill. then Condition C2 can be satisfied.
rM.
then. by Rouche Theorel [31}. the equation (33) has ng roots inside r M' Further. if lie can sholl that
(iii) The Proof of Condition C3 Since kp is bounded. for SE f(-fl. ~ Ifl). one has
C2: and C3: I (s+fl)nr-l(s+f2kp/ kpo)(stf3)' IIhere
SE
I 0f(S)t(kp/kpo)Nf(S)}Og(s) I - '1J }!ftrtngf3' ). I [Ng(s)-Og(s)}Nf(S)kplk po I -0[f2 nr ngf3' J.
r f :=r(-fl. ~ Ifl) u r(-Af2 . Bf2) vr (-f3. ~3f3)' If for any possible dynalical variations lIithin plant subject to AssuIPtion 3. we can show that
f2f3').
and
Proof. It is easy to see that. IIhen f i (i £ {3J) are suffiCiently large and f itl is luch greater than fi (i £ {Z.}). r M• r(-fl'~lfl)' r(-Af2.Bf2) and rc-f3.~3f3) are pairllise disjoint. Let
Cl:
(45) (46)
ai-I-nr-iCnr_I[flnr-ifrf3'). i£ {nr.}. bj_I-,_jtIC,[f3'-Jt). j d .fJ.
I (stfl)nr-l(stf2kp/kpo)(stfl)' I
I > I v(s) I. s £ r f.
- ~ lnr-lkplkpo[flnr- f2f3'
194
J.
and
Proof. FrOl (47) and (9), it follows that (sn+sn-l)E=(sn.sn-l)Wer(s)K =(sn+sn-l)WI(S)(ll(s»)-l(kp/kpo)Ng(s)-Dg(s»)Lr(s)Hr(s).
I v(s) I -0(flnr-1f2f3' ). Hence,
~atisfied
for
SE
r(-f 1! E 1f1)'
Condition C3 can
be
if fi (iE{V) are sufficiently lar,e and fH1
IS luch greater than f · (i dID. The cases where SE r (-Af2,Bf2) and SE r(-}3, E3f3) can be proved sililarly. Q.E.D. ReRrk 4. To assure Condition C2 satisfied, the relations (37) and (3B), hence the degree relationships (13) have to be satisfied. Tbe Case of Scbeae 2 FrOl (4), (29), (30) and (20), tbe characteristic equa tion of the closed-loop destcned by Scheae 2 is ,iven by
FrOl the proo! ~f Theorel 1, it is known that when f i (i E {3J) .are suffICIently large and f)+l is luch greater than fj (Idtl),ll(s) has ,(=a (Hr(s»)) roots inside r(-f3' ~3f3), nr-l (~a(Lr(s)]) roots inside r(-fj, Elfl), one root Inside r (-Af2' Bf2), and n&.( ~ a (kplkpo)N,(s)Dg(s)]) roots inside r" . Let -Af2+B ~ 2f2 denote the root inside r(-Af2,Bf2), where -1
Ng(s):=Np(s)DI(s), Dg(s):=Dp(s)HI(s). Obvious!)', this equation has the sue fOri as the equation (33). Since
where D: =A-B E 2( >0). By Parsevel theorea, S 00 (e(n)(t)+e(n-1)(t»)2dt
o =(21t)-1 S +00 I (jw)nE(jw)+(jw)n-lE(jw) 12dw -00 ~C121t-l S 00(Df2)2+w2)-ldw o =C 12( 2Df 2)-1 . In the other hand,
Ng(s) E PI (s,lp+nlJ. D,(s) E P I1s,Ip+n.), the robust stability of the closed-loop systea desi,ned by Scheae 2 can be proven in the sue way as Scheae 1. We shall only prove the robust steady-state property and the robust transient property of the closed-loop systel designed by Schele 1. The systea desi,ned by Scheae 2 can sililarly be shown to have the sue properties .
S t(e(n)(T)+e(n-1)(T)]2dT
o
=(e(n-1)(t)]4 S t{ (e(n)( T) )2+ (e(n-l)( T) )2}d T . o Hence, when f2 is sufficiently large, the eqs . (50) and (51) can be satIsfied. Q.E.D.
4.2 Robust Steady-state Property Now, we investigate the robust steady-state property, i . e., the robust aSYIPtotical track in, property of the closed-loop systea. Let E denote the output error, i.e., E:=Y-Y I . It can be shown that for Scheae 1 the transfer function frOl the reference input R to the output error E, denoted by Wer(s), is given by Wer(s)=WI(s) (ll (s) ]-1 (kp/kpo)Ng(s)-Dg(s) )Dr(s)Hr(s),
Reaark 5. Frol Theorel 2, one can see that, if the initial conditions of the plant and the reference lodel are the sale in the sense of eqn.(49), and if the relative degree of the reference .odel is greater than or equal to one, then the sup I e(t) I can be lade as sull as desired, that is, the robustness of the transient property can be guaranteed.
(47)
where II (s) is, as defined in eqn. (33), the characteristic polynolial of the closed-loop systea. If the paraleters fi (i dJJ) are suitably designed such that for all possible dynalical variations within the plant, the robust stability of the closed-loop systel is guaranteed, then E (=Wer(s)R) is a strictly proper and as)'lptotically stable ratIonal function. Therefore,
5. DESIGN PROCEDURES
Two design procedures are sUllarized in this chapter.
lilt-+ooe(t)=O, (48) that is, the robust as)'lptotical tracking can be achieved. 4.3 Robust Transient Property
The Loo-norl of the output error transfer function Wires) is used as a leasure of the robustness of the c osed-loop perforlance in the literature. However, it is well known that, when I Wet:(s) I Loo is slall, we can only say that, for a ,iven reference input r(t) with bounded energy, the energy of the output error e(t) is slall. The condition that the energy of the reference input r(t) is bounded is too restrictive. Furtherlore, latheaatically, there exists a sequence of tile functions whose L2-norl converges to zero, but their supreaUl grows boundlessly. FrOl this viewpoint, instead of I We (s) I Loo, we prefer to consider directly the proper~ies of the supreaUl of the output error e(t). Theorea 2. Suppose the assuIPtions .ade in Theorea 1 hold, and suppose nl-ll~l. For any intecer k, O~k~nl -I.-I. and any constant E 0>0, if (49) e(i)(t) I t~O =0, O~ in, then (50) SUPt I e(i)(t) I
195
Sch!!lle 1: I!Akl. (1) NOlinal plant: kpp, Npo(s), DpQ(s); Reference lodel: kl' NI(S), D,ts); Reference input: R=Lr(s)/Dr(s) . (2) Paraleter variation rallies : Sk, SN, SD; Degree bound: 1 0 , no . (3) Robustness leasure of the transient property: EO. ~ Low-order lodel latchilll controller desicn (1) Calculate the reference plant k oNo(s)/D (s). (2) Design a lodel latching controlYer for ~he reference plant via (11) and (12) with the decree relationships (13) satisfied. ~ Robust cOlpensator design (1) Construct a robust filter F(s) as (20)-(24). (2) Find constants a and b, a>b>O such that bE,,/(1-EI)' a 2>(1+ E I)!b, a3>a/(1-E3); (b) and when fl~ a I, f2=a2f1 and f3=a3f2' then, for all possible plant dynuic varIations in S k' S N and SO' conditions Cl, C2 and C3 are satisfied and sup I e( t) I
IQ' no, ns' sign(kp)' (2) Reference loael: kl' NI(S), 0,(5); Reference input: R=L (s)/O (s). (3) Robustness leasure of the b-ansient property: E o. ~ Construct the input as (29) and (30). ~ The design of the robust filter F(a) ia the sSle · aa Schele I after the following changes: kp/kpo -+ kp, Ng(s):=Np(s)OI(s)H(S) -+ Ng(s):=Np(s)o.(s), 0g(s):=Op(s)[Ou(s)-Nu(s))-kpNp(s)Ny(S) Q.E.O. -+0g(s):=Op(S)HI(S). Fig.l. The output error e(t) and its energy.
6. AN EXAMPLE AND SIMULATION RESULTS
REFERENCES
As an eXSlple, consider a plant, the transfer function of which in nOlinal case is Gpo (S)=(S+I)(s+2)[(s-3)(S-4)(S+5))-I, and in actual case is Gp(s)=kpllp(s)/Op(s), where (J [Dp(s)) ~ 4=no and ns= 1. Let
(1)
[2 J (3)
Np(s)=s3+~s2+bIS+bo,
[4J [5 J
Dp(s)=s4+a3s3+a2s2+als+ao' The paraleters of the plant vary in step-wise fashion in the following ranges: Sk: kpE [0.5, 2) SN: b2E [3, 10), bl E [17, 25), bo E [I, 15) SD: a3E [-5,5), a2E [-30,25), alE [-lOO, 120), aOE [-180,400) The reference lode 1 is given by YI =[5/(s+5))R, R=I/s. Let E 0=0.01. . We can use the reference plant of the fOri
[8J
Go(s)=1/(s-3), for Schele I, and Go(s)=I/(s+5), for Schele 2. The controllers are constructed as follows.
[12J
Schele I:
[S) [7 )
[9 )
(10) (11)
[13 J
U=-8Y+5R+V, V=-[f/(s+f))[(s-3)Y-U);
[14J
[l5J
U=5R+V, V=-[f/(s+f))[(s+5)Y-UJ. Both controllers designed by Schele I and Schele 2 are of first order. It can be shown that when f;&:700, conditions Cl, C2 and C3 can be satisfied. The silulation is done for an actual plant of order 4 with the transfer function Gp(S)=1.5(s3+7.5s 2+18s+14)(s4+3s 3-24s 2-82s+2S0)-I. Fig.1 shows the properties of the output error e(t) and its energy, which are very slall for both scheles:
Schele 2:
[l6J [17 J [ 18J [I9J (20)
sup I e(t) I < 0.005,
[21J
7. CONCLUSION
[22J
Two lethods have been presented for designing loworder robust lode 1 latching controllers for SISO linilul phase plants satisfying the invariant properties. These lethods have the following features: (i) The paraleters and order of the plant can vary in any but bounded and known ranges. Hence, large dynalic variations within the plant are allowed. (i i) The order of a controller designed by these lethods is deterlined by the relative degree of the plant for given reference lodel and reference input, independent of the order of the plant. In sOle cases, the controller order can even be lower than that of a usual lodel latching controller. (iii) The controllers designed by Schele 2 have sSle structure for plants with sale relative degrees for given reference lodel and reference input. (iv) Robust stability, robust steady-state property and robust transient property can be guaranteed silultaneously, without laking any tradeof!. These lethods can be extended to lultivariable cases [32J.
[23J [24 J [25) [26J [27J [28J [29J [30J [311
[32J
196
I. Horowitz: Synthesis of feedback systels. New York, Acadelic Press(19S3). H.H. Rosenbrock: COIPuter-aided control systel design. Acadelic Press(1974). A.G.J.MacFarlane:AutOlatica, pp.455-492 (1972). P.Q. Mayne: Autolatica, pp.201-207 (1973). R.W. Brockett and C.I. Byrnes: IEEE Trans. Autolat. Contr., vol.AC-2S, pp.271-284 (1981). B.D.O. Anderson and J.B. Moore: Linear optilal control. Prentice-Hall, Inc.(1971). M.G. Safonov and M. Athans: IEEE Trans. Autolat. Contr., vol.AC-22, pp. 173-179 (1977). J.C. Doyle: IEEE Trans Autolat. Contr., vol.AC-23, pp.756-757 (1978). J.C. Doyle and G. Stein: IEEE Trans. AutOlat.Contr., vol.AC-24, pp.607-611 (1979). J.C. Doyle and G. Stein: IEEE Trans. AutOlat.Contr., vol.AC-26, pp.4-16 (1981). J.B. Moore and L. Xia: IEEE Trans. AutOlat.Contr., vol.AC-32, pp.512-517 (1987). E. Soroka and U. Shaked: IEEE Trans. AutOlat.Contr., vol.AC-29, pp.664-664 (1984). G.Zales: IEEE Trans. AutOlat. Contr., vol.AC-26, pp.301-320 (1981). B.A. Francis and G. Zales: IEEE Trans. Autolat. Contr.,vol.AC-29, pp.9-16 (1984). J.C. Doyle: Lecture notes in advances in lultivariable control. ONR/HoneyWell Workship, Minneapolis (1984) . K. Glover: Int.J.Control,vol.39,pp.1115-1193 (1984). J.W. Helton: IEEE Trans. AutOlat. Contr., vol.AC-30, pp.1154-1170 (1985). D.J.N. Lilebeer and Y.S Hung: SIAM J. Control Opt., vol.25, pp.1457-1493 (1987). D.J.N. Lilebeer and G.D. Halikias: SIAM J. Control, Opt., vol.26, pp.646-677 (1988). J.Doyle,K.Glover,P.Khargonekar and B. Francis:Statespace solutions to standard H2 and Hco control problels. to appear (1989). R.W. Scott and B.D.O. Anderson: AutOlatica, vol.14, pp.481-492 (1978). P.V. Kokotovic: SIAM Review, vo1.26, pp.501-550 (1984) . E.J. Davison and I.J. Ferquson: IEEE Trans. Autolat. Contr., vol.AC-26, pp.93-110 (1981). E.A. Jonckheere and L.M. Silverlan: IEEE Trans. Autolat. Contr., vol.AC-28, pp.953-964 (1983). B.C. Moore: IEEE Trans. Autolat. Contr.,vol.AC-26, pp.17-32 (1981). D.C. Huland and D.S. Bernstein: IEEE Trans. Autolat.Contr.,vol.AC-30,pp.1201-1211 (1985). Y. Liu and B.D.O. Anderson: Int.J. Control. vo1.44, pp.507-531 (1986). R.J. Ober: Int.J. Control, vo1.46, pp.643-670 (1987) . S.Y. Kung and D.W. Liu: IEEE Trans. AutOlat.Contr., vol.AC-26, pp.832-B52 (1981). W.A. Wolovich: Linear lultivariable systels. Springer-Verley (1974). W.Rudin: Real and cOlplex analysis, McGraw-Hill, Inc. (1974). Y.-S.Zhong: Low-order robust lodel laching controller design. Ph.D. thesis, Dept. of Electrical Engineering, Hokkaido University, Japan (1988).
LOW-ORDER ROBUST MODEL MATCHING CONTROLLER DESIGN FOR SI SO PLANTS Yi-Sheng Zhong*, T. Eisaka** and R. Tagawa** *Dept. of Autolllatioll , T ;illgil lla Ullil 'enit)' . Beijillg. PR C ** Dept. of Elect rica l E llgill eerillg, H okkaido 1.; llil'e 1'5 it)" Sapporo , j apall
Two design lethods of low-order robust lodel latching controllers for SISO plants with paraleter and order variations are presented . It is assuled that the relative degree and the sign of the gain constant of the transfer function of the plant are fixed, the zeros of the plant are aSYIPtotically stable and the coefficients and the degrees of the denolinator and nUlerator polynOlials of the plant transfer function vary in step-wise fashion in bounded and known ranges. A low-order robust lodel latching controller consists of an ordinary lodel latching controller and a robust cOlpensator. The lodel latching controller is designed by the usual lethod for a reduced-order "reference plant" constructed suitably. The nOlinal and actual plants are considered as those the reference plant changed into. The robust cOlpensator is designed to reduce the influence of the plant characteristic variations frol the reference plant. It is shown that, by suitably designing the robust cOIPensator, one can ensure that the closed-loop systel with a low-order robust lodel latching controller has robust stability, robust steady-state and robust transient properties silultaneously. For given reference lodel and reference input, the order of a controller designed by these lethods is deterlined by the relative degree of the plant, in spite of the order of the plant.
~.
~.
Control systel design; linear systel; robust control; stability; tracking systels. fixed, where the relative degree is defined as the difference of the degrees of the denolinator and the nUlerator polynolials of the plant transfer function. The paraleters and the order of the plant, subject to the incan vary in variant properties (i), (ii) and (iii), step-wise fashion in any but bounded and known ranges. Two design lethods will be presented frol different starting-points. For given reference lodel and reference input, the order of a controller designed by these lethods is deterlined by the relative degree of the plant, independent of the plant order. It is shown that, the robust properties, i.e., robust stability, robust steady-state property and robust transient property of the closed-loop systel can be guaranteed silultaneously. This paper is organized as follows. In the next chapter, the design problel is fOrlulated, and assulptions on the plant and on the reference lode I are lade. Two design lethods are presented in Chapter 3. In Chapter 4, the robustness of the closed-loop systel is proved . Design procedures are sUllarized in Chapter 5, and an exalPle and silulation results are shown in Chapter 6.
1. INTRODUCTION
Much attention has been paid to the design problels of robust control SystelS, and various design procedures have been proposed under different conditions for different types of systels . We focus our attention on linear tile-invariant systels. The classical frequency-dOlain lethods are powerful tools for designing robust controllers for single-input single-output systels [lJ, and are extended to analyze and synthesize lulti-input lultioutput Systels [2,3,4,5J. Control systels designed by LQG techniques have at least ±60· phase nrgin, infinite increasing gain largin and 50 percent gain reduction tolerance [6, 7J. Such robustness will be lost when state observers are used in the ilplelentation [8J. Loop recovery techniques can be used to ilProve the robustness of LQG control systels with observers [9,10,l1J. Unfortunately, LQG control systels have poor robustness to plant paraleter variations [12J . The HOO-optilal theory, proposed by G. Zales, is suit to deal with the design problels of robust control Systels for plants with uncertainties and/or disturbances belonging to prescribed set [13-17J. However, these lethods generally lead to high order controllers for high order plants. Controllers designed by LQG or HOO-optilal techniques have orders roughly the sale as the orders of the plants [6, 18-20J. There are several lethods to obtain low-order controllers. One can solve the linilal design problel to get low-order controllers [21J. One can use singular perturbation techniques [22J and paraleter optilization lethods [23J or apply sOle invariant properties [24J to derive low-order controllers. Model reduction techniques [25-29J can also be used to get low-order controllers. In our opinion, however, the existing results on the control systel designs for linilul phase plants are restrictive both in the robustness of the closed-loop properties and in the order reduction of the controllers. In this paper, we deal with the design problels of low-order robust lodel latching controllers for singleinput single-output linear tile-invariant plants. It is assuled that the plant to be controlled has the invariant properties: (i) its gain constant has a fixed sign, (ii) it is of linilul phase , and (iii) its relative degree is
tMill.Jm..
{oJ:
{I,2, ... ,n}
R: The set of all real nUlbers c: The set of all cOlplex nUlbers C o: {x : x e C, Re(x)
The single-input single-output (SISO) plant in nOlinal case is assUled to be described by the following equa tion. 191
Y=Gpo(s)U, Gpo(s)=kpoNpo(s)/Opo(s), (1) where Y and U are the output and the input of the plant respectively, kpo is a nonzero constant,Npo(s) E Pl [S,lpoJ and 0po(s) E P l[s,npo ). that is, Npo(s) and 0po(s) are Ionic polynOlials of degree lpo and degree npo respec t i ve ly . On the nOlinal plant (l), the following assuIPtions are aade. AsSUlption 1. (All) The degree npo and the relative degree nso := Rpo -lpO are known . The gain constant kpo and the coefficients of the polynOlials Npo(s) and Dpo(s) are known. The gain constant kpo is positive. (A12) Npo(s) is a Hurwitz polynolial, i.e, all the zeros of Npo(s) are in C_. (A13) Npo(s) and Dpo(s) are coprile. Obviously, the assUlption that kpo is positive causes no loss of generality . When kpo is negative, we can change the sign of the input. The reference lodel which specifies the desired control properties of the closed-loop systel is given by (2)
where YI is the output of the reference lodel and R is the reference input (external input) which is piecewise continuous, NI(s)E Pl[s, .. J and DI(s)E Ptls,IltJ. The reference lodel (2) is assUled to satIsfy the following assulptions . AsSUlption 2. (A2l) DI(s) is a Hurwitz polynolial . (3) (A22) p :=(nl-II)-nso~O. Under AssuIPtion 1 and 2, one can apply the usual lethods [30J to design a controller for the nOlinal plant such that the transfer function frOl R to Y of the closed-loop systel is identical with that of the reference lodel, i.e., WI(s), after having canceled sOle COlIon stable factors of its nUlerator and denOlinator polynolia!s. However, if a controller such designed is applied to the "actual" plant with dynaaic characteristics different frol the nOlinal plant, the control properties of the closed-loop systea lay becOle quite different frOl those specified by the reference lodel, and the closed-loop systea lay even lose stability. To deal with these problels, we lake sOle assuIPtions on the actual plant. Suppose the actual plant with paraleters and order having changed in step-wise fashion frol their nOlinal values is represented by
Y=Gp(s)U, Gp(s)=kpNp(s )/Dp(s), (4b) where Np(s) E P l[S,lpJ and Dp(s) E P ds,np]. The goal of this paper is to present two design lethods of low-order robust controllers for plants satisfying AssuIPtion 1 and 3 and reference lodels satisfYing AssuIPti on 2. 3. CONTROLLER DESIGN
We shall present two different design lethods. Scheme 1
First, we show a design lethod which applies the prior inforaation on the nOlinal plant. Let 1, if n~ is an odd posi ti ve integer and ,s 0:= [ 0po~s) has no real roots, (5) 0, else. It is easy to prove the following fact.
Leila 1. The transfer function Gpo(s) can always be separa ted as (6) Gpo(s)=Go(s)+Nro(s)/Gpo(s), where (i) Go(s) :=kpoNo(s)/Oo(s), (7a) (iD Dpo(s)=Do(s)Dl(S), DO(S)E Pl[s,n s+<5'0). (7b) (iii) a [Opo(s)J-a [Nro(s)J >ns ; (8) (iv) -1, if <5' 0=0, No(s):= \ _ s+c, If <5' 0=1, c is a positive constant such that Do(s) and No(s) are coprile. Reurk 1. (i) The constant c can allost be anY positive constant. In fact, it is only required that the c do not belong to the set {a: ID (-a)=O,aE R}. (ii) When ns is an od3 integer and 0po(s) has no real roots, to ensure all the coefficients of Do(s) being real,we have to set a [D o(s)J=n s +<5' o.To set a [N o(s)J=<5' 0 is to guarantee that the relative degree of No(s)/Do(s) is equal to that of Gpo(s). For the convenience of statelent, in the sequel, we call the systel with the transfer function Go(s) reference plant. Suppose the reference input is of the fora
Y=Gp(s)U, Gp(s)=N(s)/D(s), (4a) which satisfies the follOWing assuIPtions. AsSUIPtion 3. (A3l) N(s) E P [S,lpJ and D(s) E P [s,npJ, Ip:iiinl!; Ip:iiilo,np :iiino,where 10 and no are known and bounded positive integers. (A32) The coefficients of N(s) and Des) are bounded and vary in step-wise fashion in the known ranges S N and So respectively . (A33) N(s) 15 a Hurwitz polynOlial for all its coefficients in S N. (A34) There exists a fixed positive integer ns (called relative degree) , for all the coefficients of N(s) and D(s) in SN and SD respectively, such that lils-+ooGp(s)sns =:kp, w~ere kp (called gain constant) is a nonzero positIve constant taking value in the known and bounded range S k' and ns=npo-lpo=np-Ip. The assulptions (A3l) and (A34) ilply that the grees of N(s) and D(s) can vary, but their bounds known and their relative degree is fixed. The ranges and So are only requ ired bounded and knOllll. So, paraleter variations can be very large.
The equation (4a) can be rewritten as
deare SN the
192
R=Lr(s)/Or(s), where Dr(s)EP1{s,n J and a [Lr(s)J
(9)
U=[Ou(s)J-l[Nu(s)U+Ny(s)Y+krNr(s)R+VJ, (10) where V is an auxiliary input which will be given later; kr=k,/kpQ; Du(s) is a Ionic Hurwitz polynolial; Nu(s), Ny(s) and Nr(s) are polynOlials deterained by the fOllowing equations. Nr(s)=H(s)N~(s), (11) Do(s)Nu(s)+kpoNo(s)Ny(s)=Do(s)Du(s)-DI(s)H(s)No(s) ,(12) where H(s) is a Hurwitz Ionic polynolial of appropriate degree such that the follOWing degree relationships are satisfied.
Degree Relationship a [Dues) J=nl-n s +a [H(s) J; (13a) a [Du(s)J=lax{O,ns+,s o-l,na-ns} ; (l3b) a [Nu(s) J :iii a [Ou(s) J-l; (l3c) a [Ny(s)J:iii l" a [Du(s)J-l, ns=O, (13d) a [Du(s»). ns>O. The above degree relationships can always be satisfied by suitably choosing the degree of H(s). One can see that, if we set the auxiliary input V=O, then the control law given by eqn .(lO) is just an exact lodel latching control law for the reference plant. Since No(s) and Do(s) are coprile, the equation (12) has a solution for
Nu(s) and Ny(s). The actual plant can be represented in teras of the paraleters of the reference plant as
plant Go(s) frOl the nOlinal plant; any rational function can be chosen as our starting-point of design on a superplane in the rational function space specified by the invariant properties (i), (ii) and (iii). That is, we can use any rational function satisfying the Invariant Property as the "reference plant". Thus, we can choose
(14) °o(s)Y=kpoNQ(s)U+q, (15) q=[Oo(s)-Op{s)]Y-[kpoNo(s)-kpNp(s)]U. SUbstituting the U given by eqn.(IO) into (14), and using (11) and (12), one has
(26)
Y=WI(s)R+[OI(s)H(s)No(s)]-I{kpoNo(s)V+[Du(s)-Nu(s)]q}. (16) Here, we can consider the actual plant (4) to be one the reference plant changed into after dynaaics (paraaeters and degrees of its transfer function) having varied. The variable q given by eqn.(I5), called equivalent djstur~, represents the influence of the dynalical variations on the control properties of the closed-loop systel. We want to design the auxiliary input V to elilinate or reduce the influence of the equivalent disturbance q. If we set the auxiliary input V as
where H.(s) is any Ionic Hurwitz polynOlial of degree nl -n s ' and o.(s) is the denOlinator of the transfer function of the reference lodel (see eqn. (21». Obviously, the rational function "-(S)/OI(S) satisfies the invariant properties (i),(ii) and (iii). The description of the actual plant in terls of the polynolials "-Cs) and 0l(s) then becOles 01(s)Y=HI(s)U+q, (27) q=[OI(s)-Op(s)]Y-[HI(s)-kpNp(s)]U. (28) The control input U can be constructed as
V=-[kpoNo(s)]-I[Ou(s)-Nu(s)]q, (17) then, exact lodel latching is achieved in spite of the dynalic variations within the plant. However, the control law given by eqs.(IO) and (17) cannot be ilPlelented. So, a filter F(s), called robust filter, is introduced, and the auxiliary input V is constructed as (18) V=-F(s)Z, (19) Z=[kpoNo(s)]-I[Ou(S)-Nu(s)]q. The robust filter F(s) is of the fori:
(29) (30) where F(s) is a robust filter given by equations (20-24).
U=[H.(s)]-I[k,NI(S)R-F(s)q],
q=OI~s)Y-HI(s)U,
Since the polynolial H (s) can be chosen arbitrarily in the set of Ionic Hurwi'z polynOlials of degree nl-ns' the paraaeters to be deterlined of the controller described by equations (29) and (30) are only the fi (iE (~}). For given reference lodel WI(s) and reference input R, the structure of the controller is deterlined only by the relative degree ns ' which decides the degree of the polynolial HI(s). The choice of the paraaeters fi (iE (~) depends on the ranges of the dynalical (i.e., paraleter and degree) variations within the plant. The Order of the Controller Now, we cOlpare the orders, denoted by n, of the controllers designed by the lethods presented a~ove. For Schele I, ns=O, nr+nl' (31) nc= [ nr -1+.ax{2ns +o o-1.n I L ns>O; and for Schele 2, ns=O; nr+nl' (32) nc= { nr +n l -1. ns>O. For exaaple, in the case where ns=I, R=ro/s, roE R, WI(s)= al/(s+a l ), a. ER and al>O, 1+ 0 0 , for Schele I, n- { c- 1. for Schele 2, in spite of the order of the plant!
F(s)=Nf(s)/Of(s), (20) (21) °f(S)=(s+fI)nr-I(s+f2)(s+f3) , , t =lax{O,ns-I}, (22) (23) f3 »f2 »fI >0; (24a) Nf(S)=Of(s)-Or(s)Hr(s), Nf(s)E P[s.nr-I]. Hr(S)E PI[s, t]. (24b) where Ores) is the denOlinator of the Laplace transforl of the reference input (see eqn.(9». The sYlbol M»~" in eqn. (23) leans "Iuch greater" . FrOl equations (19), (15), (4) and (12), one can have Z=[OI(s)H(s)/ kpo+Ny(s)]Y-[Ou(s)-Nu(s)]U. (25) We call the part of the controller described by eqs. (18), (20) and (25) robust cQlpensator. Relark 2. (i) It is easy to see that the whole controller represented by equations (10). (18), (20) and (25) is realizable. (ii) The order of this controller is deterlined by the degrees of 01(S) and 0r(s), and the relative degree ns of the transfer function of the plant. and is independent of the degree of 0po(s) or 0p(s), i.e., the order of the nOlinal plant or the actual plant.
Reeark 3.
(i) We can see that the order of a controller designed by lethods presented above can even be lower than that of an exact lodel latching controller designed by the usual lethods [30] in the following cases:
Scheme 2
(a) ns=O and np>nr+nl+I; (b) ns>O and np>n r +lax{2n s-o o-I,n l } for Schele I, np>nr+nl for Schele 2.
The lain idea of the design lethod stated above is that, a lodel latching controller is first designed for the reference plant Go(s) derived frOl the nOlinal plant, the influence of the "dynalical variations" of the plant frOl the reference plant is represented by the equivalent disturbance q, then a robust cOlpensator is designed to reduce the influence of the equivalent disturbance q. The starting-point of this design lethod is the reference plant . The reference plant has the following properties. which are the sale as those of the plant. Invariant Property (i) The gain constant is positive. (i i) All the zeros are in C _. (iii ) The relative degree is equal to ns' FrOl the analysis on the properties of the closedloop Systels with a controller designed by the lethod stated above, we see that the locations of the zeros and poles of the reference plant is not ilportant, but the invariant properties (i), (ii) and (iii) are crucial. So, it has no need of deriving the reduced-order reference
(ii) Choosing a lower order reference lodel can lead to a lower order controller. However. since n.~ns' the order of a controller designed by the lethods presented above is finally deterlined by the relative degree ns of the plant for a given reference input.
4. ROBUSTNESS
In this chapter. we shall investigate the robust control properties, i.e., robust stability, robust steadystate property and robust transient property, of the closed-loop systel conSisting of the plant (4) and a controller designed by the lethods presented in the last chapter. 4.1 Robust Stability
A lella is first introduced. 193
r.e.a
2.
Let
v(s):=Of(S)+(kplkpo -I)Nf(S)-(s+fl)nr-l(s+f2 kplkpo)(S+t3) , =(l-kplkpo)[f2(S+fl)nr-l(stf3)' -Nf(s»). (41)
h(s)=sn+hn_ISn-I+ ... +hIS+ho. lIhere hi_I's (i£{n}) are real constants. Then. aax[ I a I : h(a)=O.
then. the equation (36). hence (33) has nr-I roots inside r(-fl. ~ Ifl). one root inside r(-Af2.Bt2) and , roots inside r(-f 3 • ~3f3)' Since r M• r(-fl. ~ Ifl)' r(-Af2. Bf2) and r(-t3' ~3f3) are pairllise disjoint. if conditions Cl. C2 and C3 are satisfied. then the equation (33) has ngtnf roots inSIde r M and r Obviously. all the roots are in C_. Hence. the proo of this theorel is done if lie can show that for any possible dynalical variations lIithin the plant. if the fi (i £ {J.}) are sufficiently large and fi+1 IS luch greater than fi (i £ {Z). conditions Cl. C2 and C3 can be satisfied. In the sequel. lie use the expression" a - r [,8)" to denote the fact that lil,B-+coa I ,8=r .lIhere a >0. ,8>0. r ;;: O. The condition (23) is thus expressed as
C }
f'
The case of Schelle 1 FrOl (4). (10-12). (17). (18) and (25). it follows that the characteristic equation of the closed-loop systel designed by Schele I is the following. ~(s):=(kplkpo)Nf(S)Ng(s)+[Of(S)-Nf(S)}Og(s)=O.
(33)
Ng(s):=Np(s)OI(s)H(s). Og( s) : =Op(s) [Ou(s)-Nu(s) }-kpNp(s)Ny (s).
(42)
Let
IIhere (34) (35)
(43) (44)
Nf(s)=anr_ISnr-I+ ... +als+aO' Hr(s)=s' +b, -Is' -1+ ... +bls+b o '
The equation (33) can be rellritten as
FrOl (21) and (24). it can be shown that [Of(s)+(kp/kpo-I)Nf(s)}Og(s)+[Ng(s)-Og(s)}(kp/kpo)Nf(s)=0. (36) FrOl eqn.(13). one has and
11 [N g (s)}=11 [Og(s)} (=:ng ).
(37)
where ICn:=n! / [(n-I)!I!}.
Ng(s)£ P}ls.ngl.
(38)
(i) The Proof of Condition Cl FrOl the definition of f M• one can see that there exists a nonzero positive constant '1J such that for s £ rM. I Ng(s) I ;;: '1J ;>! O.
0g(s) £ PI[s.ng}'
Let nf:=11 [Of(s)}=nrt,. Then. the equation (33) has nf+ ng roots. Since Ng(s) is a HUrllitz polynOlial and the range of the coeffIcients of Np(s) is known. by Lella 2. one can deterline a positive constant ~H such that all the zeros of Ng(s) locate inside the half-circle rH given by rM:={s:
I s I :i ~J1. Re(s)=O. or I s I = ~ H. Re(s) :i0.
SE
Hence. frol (42). (45) and (46). one has that. for s£ f M• I (kp/kpo)Nf(S)Ng(s) I ;;: '1J (kp/k po ) I N~(s~ I - '1J (kp/kpo)[fl r- f2f3' J. and I [Of(S)-Nf(S)}Og(s) I -O[fl nr - I f 2f3' ).
(39)
c}.
By the prior inforaation on S k. one can also cons tants a. b £ R. a>b>O such that
deterline
here lie have used the fact that the coefficients and degrees of 0p(s). Np(s) and 0E(s) are bounded. Therefore. Condition I can be satisfied if f i (i £ {~}) are sufficiently large. and fit I is luch greater than fi (i £ {Z).
(40) Let ~ I and ~ 3 be constants in the interval (0.1). i. e .• O<~I
(ii) The Proof of Condition C2 FrOl the boundedness of the paraleters of the plant and the eqn.(37). for SE r(-fl. ~ Ifl). i.e .• s=zlfl. zl=-l+~lexp(je) (O:;oe<21t). one has
Theorel 1. For the plant described in nOlinal case by (I) and in actual case by (4). and the reference lodel given by (2). subject to AssuIPtion I. 3 and 2 respectively. if the controller represented by (10). (17). (18) and (25) is applied. and if the paraleters fi (i £ eU) are sufficiently large and f itl is luch greater than fi (i £ {Z). then. the closed-loop systel is robustly stable. and its poles locate inside r M• r(-fl' ~ If l). r(-Af 2. Bf2) and r(-f3. ~3f3) '
I 0f(S)+(kp/ kpo)Nf(S) }Og(s) I - I zll na lnr-l(kplk 0) [flnrtng-If2f3 ' }
;;:(1-~ l)ng~ lnr-I[fln~tng-
I [Ng(s)-Og(s)}Nf(S)kp/ kpo I -0 [flnrtng-lf2f3 ' }. Sililarb. for
I (kplkpo)Nf(S)Ng(s) I > I [Of(s)-Nf(S)]Og(s) I
.SE
and for
f(-A!2.Bf2). one has
SE r(-f3.~3f3).
I 0f(s)+(kp/kpo)Nf(s) }Og(s) I -
'1J
2 [f3ngt nrt ,
I [Ng(s)-Og(s)}Nf(S)kplk po I -O[f3 ng+nr +, J. the
I.
where '1J 1 and '1J 2 are nonzero positive constants independent of f i (i d~}). So. if fi (i £ {~}) are sufficiently large and fitl is luch greater than f i (i dill. then Condition C2 can be satisfied.
rM.
then. by Rouche Theorel [31}. the equation (33) has ng roots inside r M' Further. if lie can sholl that
(iii) The Proof of Condition C3 Since kp is bounded. for SE f(-fl. ~ Ifl). one has
C2: and C3: I (s+fl)nr-l(s+f2kp/ kpo)(stf3)' IIhere
SE
I 0f(S)t(kp/kpo)Nf(S)}Og(s) I - '1J }!ftrtngf3' ). I [Ng(s)-Og(s)}Nf(S)kplk po I -0[f2 nr ngf3' J.
r f :=r(-fl. ~ Ifl) u r(-Af2 . Bf2) vr (-f3. ~3f3)' If for any possible dynalical variations lIithin plant subject to AssuIPtion 3. we can show that
f2f3').
and
Proof. It is easy to see that. IIhen f i (i £ {3J) are suffiCiently large and f itl is luch greater than fi (i £ {Z.}). r M• r(-fl'~lfl)' r(-Af2.Bf2) and rc-f3.~3f3) are pairllise disjoint. Let
Cl:
(45) (46)
ai-I-nr-iCnr_I[flnr-ifrf3'). i£ {nr.}. bj_I-,_jtIC,[f3'-Jt). j d .fJ.
I (stfl)nr-l(stf2kp/kpo)(stfl)' I
I > I v(s) I. s £ r f.
- ~ lnr-lkplkpo[flnr- f2f3'
194
J.
and
Proof. FrOl (47) and (9), it follows that (sn+sn-l)E=(sn.sn-l)Wer(s)K =(sn+sn-l)WI(S)(ll(s»)-l(kp/kpo)Ng(s)-Dg(s»)Lr(s)Hr(s).
I v(s) I -0(flnr-1f2f3' ). Hence,
~atisfied
for
SE
r(-f 1! E 1f1)'
Condition C3 can
be
if fi (iE{V) are sufficiently lar,e and fH1
IS luch greater than f · (i dID. The cases where SE r (-Af2,Bf2) and SE r(-}3, E3f3) can be proved sililarly. Q.E.D. ReRrk 4. To assure Condition C2 satisfied, the relations (37) and (3B), hence the degree relationships (13) have to be satisfied. Tbe Case of Scbeae 2 FrOl (4), (29), (30) and (20), tbe characteristic equa tion of the closed-loop destcned by Scheae 2 is ,iven by
FrOl the proo! ~f Theorel 1, it is known that when f i (i E {3J) .are suffICIently large and f)+l is luch greater than fj (Idtl),ll(s) has ,(=a (Hr(s»)) roots inside r(-f3' ~3f3), nr-l (~a(Lr(s)]) roots inside r(-fj, Elfl), one root Inside r (-Af2' Bf2), and n&.( ~ a (kplkpo)N,(s)Dg(s)]) roots inside r" . Let -Af2+B ~ 2f2 denote the root inside r(-Af2,Bf2), where -1
Ng(s):=Np(s)DI(s), Dg(s):=Dp(s)HI(s). Obvious!)', this equation has the sue fOri as the equation (33). Since
where D: =A-B E 2( >0). By Parsevel theorea, S 00 (e(n)(t)+e(n-1)(t»)2dt
o =(21t)-1 S +00 I (jw)nE(jw)+(jw)n-lE(jw) 12dw -00 ~C121t-l S 00(Df2)2+w2)-ldw o =C 12( 2Df 2)-1 . In the other hand,
Ng(s) E PI (s,lp+nlJ. D,(s) E P I1s,Ip+n.), the robust stability of the closed-loop systea desi,ned by Scheae 2 can be proven in the sue way as Scheae 1. We shall only prove the robust steady-state property and the robust transient property of the closed-loop systel designed by Schele 1. The systea desi,ned by Scheae 2 can sililarly be shown to have the sue properties .
S t(e(n)(T)+e(n-1)(T)]2dT
o
=(e(n-1)(t)]4 S t{ (e(n)( T) )2+ (e(n-l)( T) )2}d T . o Hence, when f2 is sufficiently large, the eqs . (50) and (51) can be satIsfied. Q.E.D.
4.2 Robust Steady-state Property Now, we investigate the robust steady-state property, i . e., the robust aSYIPtotical track in, property of the closed-loop systea. Let E denote the output error, i.e., E:=Y-Y I . It can be shown that for Scheae 1 the transfer function frOl the reference input R to the output error E, denoted by Wer(s), is given by Wer(s)=WI(s) (ll (s) ]-1 (kp/kpo)Ng(s)-Dg(s) )Dr(s)Hr(s),
Reaark 5. Frol Theorel 2, one can see that, if the initial conditions of the plant and the reference lodel are the sale in the sense of eqn.(49), and if the relative degree of the reference .odel is greater than or equal to one, then the sup I e(t) I can be lade as sull as desired, that is, the robustness of the transient property can be guaranteed.
(47)
where II (s) is, as defined in eqn. (33), the characteristic polynolial of the closed-loop systea. If the paraleters fi (i dJJ) are suitably designed such that for all possible dynalical variations within the plant, the robust stability of the closed-loop systel is guaranteed, then E (=Wer(s)R) is a strictly proper and as)'lptotically stable ratIonal function. Therefore,
5. DESIGN PROCEDURES
Two design procedures are sUllarized in this chapter.
lilt-+ooe(t)=O, (48) that is, the robust as)'lptotical tracking can be achieved. 4.3 Robust Transient Property
The Loo-norl of the output error transfer function Wires) is used as a leasure of the robustness of the c osed-loop perforlance in the literature. However, it is well known that, when I Wet:(s) I Loo is slall, we can only say that, for a ,iven reference input r(t) with bounded energy, the energy of the output error e(t) is slall. The condition that the energy of the reference input r(t) is bounded is too restrictive. Furtherlore, latheaatically, there exists a sequence of tile functions whose L2-norl converges to zero, but their supreaUl grows boundlessly. FrOl this viewpoint, instead of I We (s) I Loo, we prefer to consider directly the proper~ies of the supreaUl of the output error e(t). Theorea 2. Suppose the assuIPtions .ade in Theorea 1 hold, and suppose nl-ll~l. For any intecer k, O~k~nl -I.-I. and any constant E 0>0, if (49) e(i)(t) I t~O =0, O~ in, then (50) SUPt I e(i)(t) I
195
Sch!!lle 1: I!Akl. (1) NOlinal plant: kpp, Npo(s), DpQ(s); Reference lodel: kl' NI(S), D,ts); Reference input: R=Lr(s)/Dr(s) . (2) Paraleter variation rallies : Sk, SN, SD; Degree bound: 1 0 , no . (3) Robustness leasure of the transient property: EO. ~ Low-order lodel latchilll controller desicn (1) Calculate the reference plant k oNo(s)/D (s). (2) Design a lodel latching controlYer for ~he reference plant via (11) and (12) with the decree relationships (13) satisfied. ~ Robust cOlpensator design (1) Construct a robust filter F(s) as (20)-(24). (2) Find constants a and b, a>b>O such that b
IQ' no, ns' sign(kp)' (2) Reference loael: kl' NI(S), 0,(5); Reference input: R=L (s)/O (s). (3) Robustness leasure of the b-ansient property: E o. ~ Construct the input as (29) and (30). ~ The design of the robust filter F(a) ia the sSle · aa Schele I after the following changes: kp/kpo -+ kp, Ng(s):=Np(s)OI(s)H(S) -+ Ng(s):=Np(s)o.(s), 0g(s):=Op(s)[Ou(s)-Nu(s))-kpNp(s)Ny(S) Q.E.O. -+0g(s):=Op(S)HI(S). Fig.l. The output error e(t) and its energy.
6. AN EXAMPLE AND SIMULATION RESULTS
REFERENCES
As an eXSlple, consider a plant, the transfer function of which in nOlinal case is Gpo (S)=(S+I)(s+2)[(s-3)(S-4)(S+5))-I, and in actual case is Gp(s)=kpllp(s)/Op(s), where (J [Dp(s)) ~ 4=no and ns= 1. Let
(1)
[2 J (3)
Np(s)=s3+~s2+bIS+bo,
[4J [5 J
Dp(s)=s4+a3s3+a2s2+als+ao' The paraleters of the plant vary in step-wise fashion in the following ranges: Sk: kpE [0.5, 2) SN: b2E [3, 10), bl E [17, 25), bo E [I, 15) SD: a3E [-5,5), a2E [-30,25), alE [-lOO, 120), aOE [-180,400) The reference lode 1 is given by YI =[5/(s+5))R, R=I/s. Let E 0=0.01. . We can use the reference plant of the fOri
[8J
Go(s)=1/(s-3), for Schele I, and Go(s)=I/(s+5), for Schele 2. The controllers are constructed as follows.
[12J
Schele I:
[S) [7 )
[9 )
(10) (11)
[13 J
U=-8Y+5R+V, V=-[f/(s+f))[(s-3)Y-U);
[14J
[l5J
U=5R+V, V=-[f/(s+f))[(s+5)Y-UJ. Both controllers designed by Schele I and Schele 2 are of first order. It can be shown that when f;&:700, conditions Cl, C2 and C3 can be satisfied. The silulation is done for an actual plant of order 4 with the transfer function Gp(S)=1.5(s3+7.5s 2+18s+14)(s4+3s 3-24s 2-82s+2S0)-I. Fig.1 shows the properties of the output error e(t) and its energy, which are very slall for both scheles:
Schele 2:
[l6J [17 J [ 18J [I9J (20)
sup I e(t) I < 0.005,
[21J
7. CONCLUSION
[22J
Two lethods have been presented for designing loworder robust lode 1 latching controllers for SISO linilul phase plants satisfying the invariant properties. These lethods have the following features: (i) The paraleters and order of the plant can vary in any but bounded and known ranges. Hence, large dynalic variations within the plant are allowed. (i i) The order of a controller designed by these lethods is deterlined by the relative degree of the plant for given reference lodel and reference input, independent of the order of the plant. In sOle cases, the controller order can even be lower than that of a usual lodel latching controller. (iii) The controllers designed by Schele 2 have sSle structure for plants with sale relative degrees for given reference lodel and reference input. (iv) Robust stability, robust steady-state property and robust transient property can be guaranteed silultaneously, without laking any tradeof!. These lethods can be extended to lultivariable cases [32J.
[23J [24 J [25) [26J [27J [28J [29J [30J [311
[32J
196
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