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A Geometric Approach to Multivariable Perfect Tracking
Copyright © 1996 IFAC 13th Triennial World Congress, San Francisco, USA
2a-07 5
A GEOMETRIC APPROACH TO MULTIVARIABLE PERFECT TRACKING G. Marro' and A. PIazzi"
, DEIS, Universita di Bologna, Viale Risorgimento 2, 1-40136 B%gna, /lI1ly (Fax: -39-51-6443073, E-mail: [email protected]) ., 011, Universita di Parma, Viale delle Scienze.I-43JOO Parma.Jlaly
Abstract: The problem of achieving tracking without ttansients of the error variable in an asymptotically robust multivariable regu1ation scbeme based on the internal model principle is considered. A supervising unit. tha1 reproduces the reference signals by suitably switching exosystems, keeps under control the states of the regulator and of a feedforward unit the purpose of which is tu generate input-correctiog pulses. It is shown tha1 this control scheme enables tracking without error if the plant is minimum-phase and awroximate tracking in the non-minimum phase case if the signal to track is known some time in advance. Keywords: Geometric approaches, multivariable feedback control systems, non-minimum phase systems, predictive control, state-space models, tracking systems.
I. INTRODUCI10N This paper presents an attempt tu combine some basic features of control systems, that received considerable and qualified attention in the control literature, by using a new control scheme which inclndes a digital processor as a supervising unit. These features are asymptotic robustness. perfect control and preaction (or noncausaI control) tu deal with nonminimum phase systems. Asymptotic robust tracking through an internal model of the exosystem was thoroughly investigated in the mid-seventies (see for instance Francis and Wonham. 1976), while perfect control and its limitations in the nonminimum phase case was the object of subsequent studies (see Kimura, 1982; Davison and Scheninger, 1987). The importance of preview or lookahead control has heen pointed out in connection with optimal tracking since the end of the sixties, and the importance of using noncausaI algorithms in connection with nonlinear system inversion and perfect output tracking has heen emphasized in recent work (Gross "et aI", 1994; Hunt "et al". 1992; Devasia, Paden, 1994; Devasia "et aI", 1994). The problem here considered is to design a controller such tha1 the plant output exactly tracks a continuous
piecewise-smooth reference signal avoiding large peaking and unboundedness in the plant input. The plant is multiinput/multi-output continuous-time, time-invariant. The general framework follows the geometric approach to linear system analysis and synthesis (Wonham, 1985; Isidori, 1985; Basile and Marro. 1992) and the type of approach (use of a supervising unit that generates a replica of the reference signal) can be considered to be an application of a previous work by the authors on elimination of tracking error ttansients when the controlled plant and regulator are subject to large parameter jumps (Marro and Piazzi, 1993). Some words on notation and recalls are in order. R stands for the field of real numbers, C for that of complex numbers, split into C _, Co and C+ (left half-plane, imaginary axis and right half-plane). Referring to a linear time-invariant system represented by the triple (A, B, C) with A, B and C being nxn, nxp and qxn respectively, V' :=maxV(A, imB,kerC) denotes the maximum (A, imB)-controlled invariant contained in kerC, and S' := minS(A,kerC,imB) the minimum (A,kerC)-cooditioned invariant containing imB. It is known that at least one F exists such tha1 (A+BF) V' CV' and at least one G such that (A + GC) S' C S' , and-that n':= V'nS' is tbe reachable set on V'; OrAlIx,/x, de-
1223
notes the spectrum of the R x R matrix A restricted 10 the quotient space X,I X•• where X" X. are A-invariants such that X. ex,: it is Irnown that the set of all invariant zeros of (A.B-;C) is Z(A.B.C) := (J(A+BF)lv"'" with F as defined above. Furthennore. the triple (A. B. C) is zero. state invertible (left-invertible) if and only if V· n imB = to}. functionally controUable (right-invertible) if and only if S·+1rerC=Rn •
2. STATEMENT OF THE PROBLEM The control scheme considered is shown in Fig. 1. It includes a standard multi variable feedback loop consisting of a plant 1:p and a regulator 1:,.. designed 10 react 10 and asymptotically reject inaccessible and unpredictable disturbances possibly applied 10 the plant and not shown in the figure and 10 track with zero steady-state error the reference signals generated by the exosystems I:.., (i= I •...• q). The plant may include an inner loop designed to imlX"'ve IX"'mptness in disturbance rejection and to reduce parameter uncertainty. The scheme also includes a supervising unit whose aim is 10 realize. if possible. perfect tracking of vector reference inpUl r(t)=(r'(t) •...• by suitably killing tracking error transients in the feedback loop.
In practice the regulator. exosystems. filters. feedforward compensator and supervising unit can he realized as a single speciaI-porpose digital compoter. However, for a neater presentation of the results. in this paper continuons-time models will be adopted The main feedback loop, including the plant, regulator and exosystems (without filters and compensator) is assumed 10 he designed with the standard geometric approach technique 10 fulfiI both the asymptotic regulation condition and the loop stability condition and that the exosystems are repeated in the regulator to achieve asymptotically robust regulation (Francis and Wonham. 1976). These prerequisites involve some geometric IX"'perties, that, as they are used in the next section for deriving a theory 10 add filters and compensator. are herein brieOy recalled. The plant. of order Rp. is assumed 10 be purely dynamic. represented by the triple (A p , Bp, Cp) and the generic i-th exosystem. of order R•• ,. is represented by the observable pair (A •. " C ••,) (i= I, ... , q), with
.
,'
A" = [
C,.,=[I
r
~.
o
-11.,0
-rli,l
0
I
...
(1)
01
El=,
Hence the overall exosystem. of order Re = n,." is represented by (A" C.). with A. and C. defined as the block-
i(t) = A 11(/) + B u(t) e(t) = E x(t)
(2)
with
L-_+++-®.J eq
.- [AI A .0 Fig. 1. The feedback system considered with exosystems. filters and compensator. 1b this end the supervising unit: i) generates an approximate replica of the reference input by cbanging. at discrete instants of time. the leading states of the exosystems I.•.,. each followed by aft/ter (i = I, ... , q). whose porpose is 10 apply sufficiently smooth signals 10 the regulation loop; ii) reads and changes the regulator stale when necessary; iii) changes the state of a fee4forward compensator when necessary. thus generating correcting Signals that are injected on the manipolable inpot of the plant
I.,.,
r..
B.-._[B,] 0 '
A3] A• •
E:= [E, E21
(3)
while the regulator equations are z(t) = N z(t) + M e(t) u(t)
= L z(t) + K e(t)
(4)
To adapt this general plant and exosystem model 10 our case. let A,=Ap. A3=O. A2=A., B,=Bp• EI=-Cp, E.=C •• so that x, ERn, with R,=Rp denotes the state of the plant, x~ E Rn, with n~ = "IS that of the exosystem, x E Rn with n=n,+n, that of the regulated system, UERP the control input, e E R' the error output and z E Rm the regulator state.
1224
The main regulation loop in Fig. I can be described as the autooomous eX/ended system
i(t) =
Ai(t)
(5)
e(t) = Ex(t) with
, [A'
A:=
+ B,KE, A. + B,KE. 0
A.
ME,
ME,
i :=[x, x. Z)T,
It := [E,
B'oL] ,
3. THE FILTERS AND THE COMPENSATOR
The overall regulation scbeme represented in Fig. I is again considered. In this section it will be shown that tracking without error of the replicated reference signal is feasible while maintaining lbe control varialJle hounded if i) the controlled plant is functionally controllable; ii) the controlled plant is strictly minimum phase. In geometric tenns the above conditions are expressed as
N
minS(A., kerC., imB.) + kerC. = Rn. Z(A., Bp, C p) <:;; C
(6)
E. 0)
If both the regulation and plant stability conditions are satisfied,there is an A-invariant W with dimension ne, contained in kerlt, extemally strictly stable and satisfying
p, ~fined as
g I~]
(8)
is the extended plant, an A-invariant. The asymptotic regulation is robust because lbe existence of such a W is not aIfected by small parameter variations in the plant and regulator if models of the exosystems are robustly included in the regulator dynamics. This is obtained in the cOlllSe of the synthesis process by a particular choice of matrix N. Owing to (5) W can be expressed in matrix tenos as
(9) Let P:=[ln 0), an nx(n+m) matrix, be the projection from the extended system to the regulated system state space. It is easily seen that V :=PW is an (A, imB)-controlled invariant contained in kerE (hence in V') that satisfies
(10) with
"P := {x :
I
o
o o.
-IP,.,
-'i'~/"-l
[In'
"P := {i : x. = O} = im
X2
= O} = im[ln, O)T
(11)
(A .. BIl is assumed to be stabilizable and (A, E) detectable. The easiest-to-handle while not significantly restrictive sufficient conditions for solvability of the regulator )X"oblem, expressed in strictly geometric tenos. are
V"'+'P=Rn Z(A .. B"E,)nu(A.)=
(12)
0
(13)
where V' := maxV(A, imB, kerE) and Z(A" B .. C,) denotes the set of all invariant zeros of lbe triple (A"B"E,) (i.e .. of the plant in this case). Note that (12) and (13) are respectively a structural condition and a spectral condition. If they hold, straightforward algorithms provide the regulator matrices K, L, M and N (Basile and Marro, 1992). In particular, if (A .. B,) is controllable and (A, E) observable, there is a solution with m= n. in which the 2n1 +n2 eigenvalues of tbe extended plant can be arbitrarily assigned.
(15)
The filters are represented by strictly stable triples (A/",B/."C / .,), of orders nI,' (i = I, ... ,q). Like the exosystems, the filler matrices are given in the observer canonical form, i.e., with the structures
(7) where
(14)
B/,'=[O 0
...
IP,.o)T, C/,,=[I
1, 0
... 0) (16)
that guarantees absence of zeros, hence relative degree The supervising unit re)X"Oduces the reference signal n by changing the first state variable of the integrator chain present in every exosystem at discrete instants of time, so that the ouput of the i-th exosystem has the (ne.' - I)-th time derivative piecewise continuous. If (14) holds, it is possible, by a suitable choice of the input function u(t) and starting from the zero state, to reproduce wilb the output of the plant any sofficientIy smooth trajectory y(t).
I."
Regulation without error is feasible if and only if the orders of the Iow-pass filters between exosystems and summing junctions meet a relative degree condition. In the SISO case this is simply expressed as nl ~p-ne+1 in tenos of the relative degree p of the plant (the difference between the degrees of denominator and numerator of the plant lransfer function) and the exosystem order ne' In lbe MIMO case considered herein the relative degree is referred to output and its computation deserves some special altention, since the plant is nol necessari1y invertible and its input is defined only in geometric tenos (knowledge of the number of inputs is nol strictly needed, since solutions of both the regulator problem and the tracking without error problem only depend on im B). The following algorithm )X"Ovides a straightforward way of computing lbe output-referred relative degrees p, (i= 1, ••. , q), without any knowledge of the
input number. Algorithm 3.1 (perfect output controllability with respect to the "-lb derivative). Let us refer to the triple (Ap, Bp, Cpl. Given a subspace Y <:;; Rq, let £ := C p' Y; the maximal (Ap, Bp)-controlled invariant F <:;; Rn. contained in £ such that the output can be driven on C p Ft h s; Y along any
l'
1225
trajectory y(t) with piecewise continuous h-th derivative for all initial states x(O) E :Ft, is the last tenD (with identity of two consecutive tenDs as the stop condition) of the sequence
V.=[
V, = max V(Ap,imBp, v,_,nW"h_' + kerCp) where Wl,h-' is defined by the recursion process Wt,. =imBp Wl,; = imBp +Ap(W,,;_,nv,_,nkerC p)
(17)
The above algorithm was the subject of one of the earliest contributions to the geometric approach (Basile and Marro, 1972). It has now heen implemented in a very compact MatJab routine, reported in Appendix A, that uses the Geometric Approach Toolbox diskette (Basile and Marro, 1992), The relative degree p, of output y, is obtained with Algorithm 3.1 by assuming y:= {y : Y. =0, k= I, ... , q, k;!i} (the i-th coordinate axis) and repeating for h= 1,2, ... until equality y = C p:Fe' is oblained. When this occurs, it follows that p, = h. Similarly to the SISO case, the order of filters that make perfect regulation possible is given by the inequality (i=I,,,.,q)
(19)
that, in addition to strict stability, expresses the only requirement for filters. If the right member of (17) is zero or less, a filter is not required. 1b include this case filters are represented with quadruples (Aj,.,B".,C".,Dj,') with the agreement that D',' = I if the other matrices are empty and D".=O otherwise. The overall filtering system can be represented by the quadruple (A" B" C" D,), where matrices A, and D, are block-diagonal built with those of the single filters, while B, and C, are built in the same way, but completed with zero columns or rows to save dimensional congruence when dynamics is absent It is possible to include filters in the mathematical model (2) by redefining the submatrices in (3) as x, :=
[:~ J,
X2:= x., A,:=
A. := A., Aa:= E, := [-Cp
[B~C.J.
[~
B,:=
C, 1, E 2 := D,C,
to an Input signal genemted by changing the leading states of the exosysterns at IzO. It is possible to obIain 0(1)=0. I 2: 0 by suitably setting at t = 0 the states of both the regulator and a feedforward compensator (A" Cc) such that the eigenstructure of A. includes those of all filters and of the invariant zeros structure of the plant.
Proof. Let x",(O+) = (0,0, ... , a;)T (i = I, ... , q) be the (18)
0'=I,.",h-l)
n".2:p.-n",+1
Theorem 3.1. Assume that (19) holds and suppose that the overall system of Fig. I, initially in the zero state, is subject
1: J
[~J (20)
so that Xl E Rl'iol. with "1 = np+n/. now denotes the state of the system "plant plus filters". Note that (A" B,) is stilI stabilizable, (A, E) detectable, and (12), (13) still hold, Furthermore, the regulator designed for the plant in the absence of filters stin satisfies both the asymptotic regulation condition and the loop stability condition, since the modes of the filters tend to zero as time approaches infinity, like those of the control loop. The feedforward compensator is represented by the observable pair (A" Cc), of order ne' The operation of the supervising unit is based on the following theorem.
column vectors of the initial states of the exosystems and denote with x.(O+) the overall exosystem initial state, a column vector having these vectors as elements. Consider the basis matrix (9) for Wreferring to the regulator problem with matrices defined by (218), i,e., with fillers modelled as an uncontrollable part of the plant The overall system initial state corresponding to tracking without b'ansients is computed as
(21) Since the relative degree condition (19) is satisfied, perfect tracking by means of a suitable control function is possible. This itnplies that state x,(O+) belongs to an (A" itnB,)-controlled invariant conlained in kerE" hence to Vi=maxV(A" imB"kerE,), since only if this is so does a corrective control function u.(I) exist that, starring from -x,(O+), mainlains the plant state trajectory on Vi, hence with no influence on the error variable. By superposition, this control function keeps the error variable at zero if applied with our actual initial state (0, x.(O+), Z(o+»T, It is easily shown that u.(I) can be generated by the compensator: let us denote by ne the dimension of Vi and by T, a n, x n. basis matrix of Vi. Let F 1 be a matrix such that (A, +B,F , ) Vi C; Vi. define T = [T, T.] with T. sucb that T is nonsingular and assume as A. the first n. x n. submatrix ofT-'(A,+B,F,)T and as C. the pxn. matrix F, T ,. The initial state x.(O+) of the compensator is then given by the first n. components of - T-' x,(o+). The internal unassignable eigenvalues of Vi are the union of the elements of Z(Ap , Bp, C p) (the invariant zeros of the plant) and of the eigenvalues of the filters, that appear as invariant zeros of (A"BloE,) since filters are not controllable. Hence the corresponding eigenstructures are repeated in Ac. D
Remark. The role of the compensator is to drive the state of the plant and filters along trajectories on the (A,+B,F, )invariant Vi, thus making it possible to reproduce in the controlled plant the effect of state feedback matrix F, for every initial state belonging to Vi. Owing to Theorem 3.1 it is possible to oblain an asymptotically stable compensator if and only if the plant is minimum phase. Hence condition (14) is sufficient to ensure zero b'aCking error of the replicated reference signal with bounded input in this case. Note that F, can and must also stabilize the internal assignable eigenvalues of Vi if the controlled plant is not invertible.
1226
",
I
P"
Fig. 2. A correcting pulse with preaction. 4. USING PREACTION WHEN THE PLANT IS NON-MINIMUM PHASE The proof of Theorem 3.1 in the previous section suggests a working algorithm f(I the supervising unit to deal with exosystems state switching at t=O. Since the overall sysrem is linear and time-invatiant. multiple switchings at different instants of time are easily handled by superposition and time shifting. Note that the regulator state C
starting from -x, .• «H.) and tending to zero as time approaches infinity. while for xe•• «H.) the compensat
(a)
lIe(t)11 ~ 0
i:
oo
(b)
as tprc~-oo
11e(t)II' dt ~ 0
as t....
(25)
~ -00
(26)
Proof. Tracking error would be zero if the state of the plant (inclnding filters) at t=tprc were x,(t.... )=T,.A•.•• ".
[Xc.u~(H.)]
(27)
but, since the contrul action starts at t.... it is zero. The error transient can be computed by considering the response to -x,(tp ,,) oftbe extended plant (that, in this case. consists of plant. fillers and regulator) with tbe regulator state z(t.... ) at zero. Let us denote with v=(a" ...0;.,) the vector of the exosystems initial states and let Cl, (> 0) be the minimal real part of the eigenvalues of Ae.u; since lie .«H.) is a > 0 exists such that linear function of v. a constant
Ii, Ilx,(t....)1I :0; Ii,.····" IIvll
(23)
(28)
Let wbere A ••• has the strictly stable eigenvalues and A,.u those strictly unstable. Without loss of generality. transformation T e is assumed to be included in T ,. so that Ac has the structure shown on the right of (23) and the initial state x,«H.) of the compensator is accordingly partitioned as Xc •• «H.). x,.u«H.). Let x, •• «H.):= T,
[X,.•(~(H.)],
x'.u«H.) :=T,
[x,.u~(H.)]
Aep '= [A, +ME! B,KE, •
B,L] E t!p '[E 1 N' .-
0]
(29)
be the matrices of the extended plant The error is given by e(t-tprc )
=E.p•A.,(.-.,,,)
[-x'gpre)],
t
2:
t .... (30)
Let us denOIe by Cl, « 0) the maximal real part of the eigenvalues of A.p : a constant !i, > 0 exists such that
lIe(t)11 :0; !i, .',1'-',,,) IIx,(t....)1I :0; Ii, 50 ."('-"'.) ." ""lIvll :0; Ii, 50 ." ""lIvll
(24) As far as x, •• «H.) is concerned. it is possible to proceed as before. generating a control action u,(t) corresponding to an output-invisible state trajectory x,(t) of the plant and filters
1227
(31)
Furthermore,
[)e{tlll' dt :5 =
I.:
lIe{tlll' dt
1.'.....00 8~ ~ .'a,(I-I,,,J .'a, ""lIvll
=
S'I\'
-20,
dt
.2a, ""lIvll
(32)
The staJement directly follows from (31) and (32),
0
=
Corollary 4.1. Let V (a., .. .aq) be the vectcr of the exosystems initial states. Then, given any number e > 0, the following holds:
465.
=> lie{t)ll:5 e, t E R => where the quantities tp,.., A •• 02 . as in the proof of Theorem 4.1.
[:00 1Ie{t)II' dt :5 5•. 5, and e are
Basile, G. and G. Marro (1992). Conrrolled and Condilioned Invarianls in Unear SySlem Theory. Prentice Hall, Englewood Cliffs, New 1ersey. Davison, EJ. and B.M. Scherzinger (1987). Perfect control of the robust servomechanism jI"Oblem. IEEE Trans. Au/DmIl/. Coner., AC-31, 689-702. Devasia, S. and B. Paden (1994). Exact output traclring for oonIinear time-varying systems. Proc. IEEE Con!. on DeciJion and COnlrol, Lake Buena VISta, CA, 23462355. Devasia, S .. B. Paden and C. Rossi (1994). Minimal transient regulation of flexible suuctures. Proceedings of the IECON '94, Bologna, Italy, 1994. Francis, B.A. and W.M. Wooham (1976). The internal model principle of control theory. AU/OIIW/ica, 12,457-
£
defined
Remark. 11Ieorem 4.1 and Corollary 4.1 refer to a single switching of the exosystems leading states from zero 10 v at t - O. In the case of multiple switchings superposition applies, as specified at the beginning of this sectioo. If the preview time of the refereoce input is infinite, the em.. transienls due 10 subsequent switchings tend to reduce since the relative preaction time interval increases (with t p .. fixed); this is also true when preview time, although not infinite, is significantly greater UUI/l the time constant of the dominant right half-plane zero. However, when the preview time, hence the preaction time for each switching, is relatively short and variations of components of" are large, the error variable, while remaining bounded, may present resonance phenomena.
Gross, E., M. Tomizulra and W. Mcssner (1994). Cancellation of discrete time unstable zeros by feedforward control. ASME J. Dynamic Syst., Measuremenl Conrr., 116,33-38. Hun~ L.R., G. Meyer and R. Su (1992). Output traclring for oonIinear systems. Symposium on Implicit and Nonlinear Systems, Ariington, 'li:xas, 314-320. Isidori, A. (1985). Nonlinear control rysltm: an imroduc· tion. Springer-Verlag, Berlin. Kimwa, H. (1982). Perfect and subperfect regulation in linear multivariable control systems. AUlomalica, 18, 125-145. Marro, G. and A. Piazzi (1993). Regulation without transienlS under large parameter jumps. Proc. 12lh !FAC World Congress, 4, 23-26. Wonham, W.M. (1985). Linear Mullivariable Control: a Geometric Approach, 3rd cd. Springer-Verlag.
APPENDIX A. A MATLAB ROUTINE FOR ALGORITHM 3.1
%
• • aaxpC.(A~B,C,.,11) • • aaxpca(&',B.C .•• l). -...xl_l perfect controllability .ubapace witb r.epect to thA l-th derlvatlv. contained in ker(.).
%
Q.Xarro •
tun~t1on
5. CONCLUSION A special cootrol scheme for multivariable systems, whOSe purpose is 10 eliminate tracking error, thus also obIaining complete noointeraction, has been discussed, and some quantitative results concerning tracking error reduction in the non-minimum phase case have been presented. The application of the same technique to multivariable discretetime systemS, in which zeros outside the unit circle are quite frequent, does not require any significant change: Algorithm 3.1 Can still he applied to detect the multivariable relative degree and the corresponding relative degree condition can be satisfied by using simple delays instead of filters.
.. %
'-30-95
"_lter (II:) 1 [ .... ,DV] •• 1 •• CM) I
Bl.laa(B,O), Cl_lt.r(e), h.O, while (nv-Dw) > 0 I h •• o
DV_n.,
Z.Blt for jj.l1 Cl1-1) Z.laa([Bl,A*lnt.(lnta(Z •• ) , Cl}l.O), _nd
REFERENCES Basile, G . and G. Marro (1972). 011 the perfect output controllability of linear dynamic systems. Ricerche di AuwmDlica (in English), 2, 1-10.
A.Pl.K~l
1_.nw] •• ! •• ,W) , _Dd % --- l •• t
1228
line of
"xpc. ---
2a-07 5
A GEOMETRIC APPROACH TO MULTIVARIABLE PERFECT TRACKING G. Marro' and A. PIazzi"
, DEIS, Universita di Bologna, Viale Risorgimento 2, 1-40136 B%gna, /lI1ly (Fax: -39-51-6443073, E-mail: [email protected]) ., 011, Universita di Parma, Viale delle Scienze.I-43JOO Parma.Jlaly
Abstract: The problem of achieving tracking without ttansients of the error variable in an asymptotically robust multivariable regu1ation scbeme based on the internal model principle is considered. A supervising unit. tha1 reproduces the reference signals by suitably switching exosystems, keeps under control the states of the regulator and of a feedforward unit the purpose of which is tu generate input-correctiog pulses. It is shown tha1 this control scheme enables tracking without error if the plant is minimum-phase and awroximate tracking in the non-minimum phase case if the signal to track is known some time in advance. Keywords: Geometric approaches, multivariable feedback control systems, non-minimum phase systems, predictive control, state-space models, tracking systems.
I. INTRODUCI10N This paper presents an attempt tu combine some basic features of control systems, that received considerable and qualified attention in the control literature, by using a new control scheme which inclndes a digital processor as a supervising unit. These features are asymptotic robustness. perfect control and preaction (or noncausaI control) tu deal with nonminimum phase systems. Asymptotic robust tracking through an internal model of the exosystem was thoroughly investigated in the mid-seventies (see for instance Francis and Wonham. 1976), while perfect control and its limitations in the nonminimum phase case was the object of subsequent studies (see Kimura, 1982; Davison and Scheninger, 1987). The importance of preview or lookahead control has heen pointed out in connection with optimal tracking since the end of the sixties, and the importance of using noncausaI algorithms in connection with nonlinear system inversion and perfect output tracking has heen emphasized in recent work (Gross "et aI", 1994; Hunt "et al". 1992; Devasia, Paden, 1994; Devasia "et aI", 1994). The problem here considered is to design a controller such tha1 the plant output exactly tracks a continuous
piecewise-smooth reference signal avoiding large peaking and unboundedness in the plant input. The plant is multiinput/multi-output continuous-time, time-invariant. The general framework follows the geometric approach to linear system analysis and synthesis (Wonham, 1985; Isidori, 1985; Basile and Marro. 1992) and the type of approach (use of a supervising unit that generates a replica of the reference signal) can be considered to be an application of a previous work by the authors on elimination of tracking error ttansients when the controlled plant and regulator are subject to large parameter jumps (Marro and Piazzi, 1993). Some words on notation and recalls are in order. R stands for the field of real numbers, C for that of complex numbers, split into C _, Co and C+ (left half-plane, imaginary axis and right half-plane). Referring to a linear time-invariant system represented by the triple (A, B, C) with A, B and C being nxn, nxp and qxn respectively, V' :=maxV(A, imB,kerC) denotes the maximum (A, imB)-controlled invariant contained in kerC, and S' := minS(A,kerC,imB) the minimum (A,kerC)-cooditioned invariant containing imB. It is known that at least one F exists such tha1 (A+BF) V' CV' and at least one G such that (A + GC) S' C S' , and-that n':= V'nS' is tbe reachable set on V'; OrAlIx,/x, de-
1223
notes the spectrum of the R x R matrix A restricted 10 the quotient space X,I X•• where X" X. are A-invariants such that X. ex,: it is Irnown that the set of all invariant zeros of (A.B-;C) is Z(A.B.C) := (J(A+BF)lv"'" with F as defined above. Furthennore. the triple (A. B. C) is zero. state invertible (left-invertible) if and only if V· n imB = to}. functionally controUable (right-invertible) if and only if S·+1rerC=Rn •
2. STATEMENT OF THE PROBLEM The control scheme considered is shown in Fig. 1. It includes a standard multi variable feedback loop consisting of a plant 1:p and a regulator 1:,.. designed 10 react 10 and asymptotically reject inaccessible and unpredictable disturbances possibly applied 10 the plant and not shown in the figure and 10 track with zero steady-state error the reference signals generated by the exosystems I:.., (i= I •...• q). The plant may include an inner loop designed to imlX"'ve IX"'mptness in disturbance rejection and to reduce parameter uncertainty. The scheme also includes a supervising unit whose aim is 10 realize. if possible. perfect tracking of vector reference inpUl r(t)=(r'(t) •...• by suitably killing tracking error transients in the feedback loop.
In practice the regulator. exosystems. filters. feedforward compensator and supervising unit can he realized as a single speciaI-porpose digital compoter. However, for a neater presentation of the results. in this paper continuons-time models will be adopted The main feedback loop, including the plant, regulator and exosystems (without filters and compensator) is assumed 10 he designed with the standard geometric approach technique 10 fulfiI both the asymptotic regulation condition and the loop stability condition and that the exosystems are repeated in the regulator to achieve asymptotically robust regulation (Francis and Wonham. 1976). These prerequisites involve some geometric IX"'perties, that, as they are used in the next section for deriving a theory 10 add filters and compensator. are herein brieOy recalled. The plant. of order Rp. is assumed 10 be purely dynamic. represented by the triple (A p , Bp, Cp) and the generic i-th exosystem. of order R•• ,. is represented by the observable pair (A •. " C ••,) (i= I, ... , q), with
.
,'
A" = [
C,.,=[I
r
~.
o
-11.,0
-rli,l
0
I
...
(1)
01
El=,
Hence the overall exosystem. of order Re = n,." is represented by (A" C.). with A. and C. defined as the block-
i(t) = A 11(/) + B u(t) e(t) = E x(t)
(2)
with
L-_+++-®.J eq
.- [AI A .0 Fig. 1. The feedback system considered with exosystems. filters and compensator. 1b this end the supervising unit: i) generates an approximate replica of the reference input by cbanging. at discrete instants of time. the leading states of the exosystems I.•.,. each followed by aft/ter (i = I, ... , q). whose porpose is 10 apply sufficiently smooth signals 10 the regulation loop; ii) reads and changes the regulator stale when necessary; iii) changes the state of a fee4forward compensator when necessary. thus generating correcting Signals that are injected on the manipolable inpot of the plant
I.,.,
r..
B.-._[B,] 0 '
A3] A• •
E:= [E, E21
(3)
while the regulator equations are z(t) = N z(t) + M e(t) u(t)
= L z(t) + K e(t)
(4)
To adapt this general plant and exosystem model 10 our case. let A,=Ap. A3=O. A2=A., B,=Bp• EI=-Cp, E.=C •• so that x, ERn, with R,=Rp denotes the state of the plant, x~ E Rn, with n~ = "IS that of the exosystem, x E Rn with n=n,+n, that of the regulated system, UERP the control input, e E R' the error output and z E Rm the regulator state.
1224
The main regulation loop in Fig. I can be described as the autooomous eX/ended system
i(t) =
Ai(t)
(5)
e(t) = Ex(t) with
, [A'
A:=
+ B,KE, A. + B,KE. 0
A.
ME,
ME,
i :=[x, x. Z)T,
It := [E,
B'oL] ,
3. THE FILTERS AND THE COMPENSATOR
The overall regulation scbeme represented in Fig. I is again considered. In this section it will be shown that tracking without error of the replicated reference signal is feasible while maintaining lbe control varialJle hounded if i) the controlled plant is functionally controllable; ii) the controlled plant is strictly minimum phase. In geometric tenns the above conditions are expressed as
N
minS(A., kerC., imB.) + kerC. = Rn. Z(A., Bp, C p) <:;; C
(6)
E. 0)
If both the regulation and plant stability conditions are satisfied,there is an A-invariant W with dimension ne, contained in kerlt, extemally strictly stable and satisfying
p, ~fined as
g I~]
(8)
is the extended plant, an A-invariant. The asymptotic regulation is robust because lbe existence of such a W is not aIfected by small parameter variations in the plant and regulator if models of the exosystems are robustly included in the regulator dynamics. This is obtained in the cOlllSe of the synthesis process by a particular choice of matrix N. Owing to (5) W can be expressed in matrix tenos as
(9) Let P:=[ln 0), an nx(n+m) matrix, be the projection from the extended system to the regulated system state space. It is easily seen that V :=PW is an (A, imB)-controlled invariant contained in kerE (hence in V') that satisfies
(10) with
"P := {x :
I
o
o o.
-IP,.,
-'i'~/"-l
[In'
"P := {i : x. = O} = im
X2
= O} = im[ln, O)T
(11)
(A .. BIl is assumed to be stabilizable and (A, E) detectable. The easiest-to-handle while not significantly restrictive sufficient conditions for solvability of the regulator )X"oblem, expressed in strictly geometric tenos. are
V"'+'P=Rn Z(A .. B"E,)nu(A.)=
(12)
0
(13)
where V' := maxV(A, imB, kerE) and Z(A" B .. C,) denotes the set of all invariant zeros of lbe triple (A"B"E,) (i.e .. of the plant in this case). Note that (12) and (13) are respectively a structural condition and a spectral condition. If they hold, straightforward algorithms provide the regulator matrices K, L, M and N (Basile and Marro, 1992). In particular, if (A .. B,) is controllable and (A, E) observable, there is a solution with m= n. in which the 2n1 +n2 eigenvalues of tbe extended plant can be arbitrarily assigned.
(15)
The filters are represented by strictly stable triples (A/",B/."C / .,), of orders nI,' (i = I, ... ,q). Like the exosystems, the filler matrices are given in the observer canonical form, i.e., with the structures
(7) where
(14)
B/,'=[O 0
...
IP,.o)T, C/,,=[I
1, 0
... 0) (16)
that guarantees absence of zeros, hence relative degree The supervising unit re)X"Oduces the reference signal n by changing the first state variable of the integrator chain present in every exosystem at discrete instants of time, so that the ouput of the i-th exosystem has the (ne.' - I)-th time derivative piecewise continuous. If (14) holds, it is possible, by a suitable choice of the input function u(t) and starting from the zero state, to reproduce wilb the output of the plant any sofficientIy smooth trajectory y(t).
I."
Regulation without error is feasible if and only if the orders of the Iow-pass filters between exosystems and summing junctions meet a relative degree condition. In the SISO case this is simply expressed as nl ~p-ne+1 in tenos of the relative degree p of the plant (the difference between the degrees of denominator and numerator of the plant lransfer function) and the exosystem order ne' In lbe MIMO case considered herein the relative degree is referred to output and its computation deserves some special altention, since the plant is nol necessari1y invertible and its input is defined only in geometric tenos (knowledge of the number of inputs is nol strictly needed, since solutions of both the regulator problem and the tracking without error problem only depend on im B). The following algorithm )X"Ovides a straightforward way of computing lbe output-referred relative degrees p, (i= 1, ••. , q), without any knowledge of the
input number. Algorithm 3.1 (perfect output controllability with respect to the "-lb derivative). Let us refer to the triple (Ap, Bp, Cpl. Given a subspace Y <:;; Rq, let £ := C p' Y; the maximal (Ap, Bp)-controlled invariant F <:;; Rn. contained in £ such that the output can be driven on C p Ft h s; Y along any
l'
1225
trajectory y(t) with piecewise continuous h-th derivative for all initial states x(O) E :Ft, is the last tenD (with identity of two consecutive tenDs as the stop condition) of the sequence
V.=[
V, = max V(Ap,imBp, v,_,nW"h_' + kerCp) where Wl,h-' is defined by the recursion process Wt,. =imBp Wl,; = imBp +Ap(W,,;_,nv,_,nkerC p)
(17)
The above algorithm was the subject of one of the earliest contributions to the geometric approach (Basile and Marro, 1972). It has now heen implemented in a very compact MatJab routine, reported in Appendix A, that uses the Geometric Approach Toolbox diskette (Basile and Marro, 1992), The relative degree p, of output y, is obtained with Algorithm 3.1 by assuming y:= {y : Y. =0, k= I, ... , q, k;!i} (the i-th coordinate axis) and repeating for h= 1,2, ... until equality y = C p:Fe' is oblained. When this occurs, it follows that p, = h. Similarly to the SISO case, the order of filters that make perfect regulation possible is given by the inequality (i=I,,,.,q)
(19)
that, in addition to strict stability, expresses the only requirement for filters. If the right member of (17) is zero or less, a filter is not required. 1b include this case filters are represented with quadruples (Aj,.,B".,C".,Dj,') with the agreement that D',' = I if the other matrices are empty and D".=O otherwise. The overall filtering system can be represented by the quadruple (A" B" C" D,), where matrices A, and D, are block-diagonal built with those of the single filters, while B, and C, are built in the same way, but completed with zero columns or rows to save dimensional congruence when dynamics is absent It is possible to include filters in the mathematical model (2) by redefining the submatrices in (3) as x, :=
[:~ J,
X2:= x., A,:=
A. := A., Aa:= E, := [-Cp
[B~C.J.
[~
B,:=
C, 1, E 2 := D,C,
to an Input signal genemted by changing the leading states of the exosysterns at IzO. It is possible to obIain 0(1)=0. I 2: 0 by suitably setting at t = 0 the states of both the regulator and a feedforward compensator (A" Cc) such that the eigenstructure of A. includes those of all filters and of the invariant zeros structure of the plant.
Proof. Let x",(O+) = (0,0, ... , a;)T (i = I, ... , q) be the (18)
0'=I,.",h-l)
n".2:p.-n",+1
Theorem 3.1. Assume that (19) holds and suppose that the overall system of Fig. I, initially in the zero state, is subject
1: J
[~J (20)
so that Xl E Rl'iol. with "1 = np+n/. now denotes the state of the system "plant plus filters". Note that (A" B,) is stilI stabilizable, (A, E) detectable, and (12), (13) still hold, Furthermore, the regulator designed for the plant in the absence of filters stin satisfies both the asymptotic regulation condition and the loop stability condition, since the modes of the filters tend to zero as time approaches infinity, like those of the control loop. The feedforward compensator is represented by the observable pair (A" Cc), of order ne' The operation of the supervising unit is based on the following theorem.
column vectors of the initial states of the exosystems and denote with x.(O+) the overall exosystem initial state, a column vector having these vectors as elements. Consider the basis matrix (9) for Wreferring to the regulator problem with matrices defined by (218), i,e., with fillers modelled as an uncontrollable part of the plant The overall system initial state corresponding to tracking without b'ansients is computed as
(21) Since the relative degree condition (19) is satisfied, perfect tracking by means of a suitable control function is possible. This itnplies that state x,(O+) belongs to an (A" itnB,)-controlled invariant conlained in kerE" hence to Vi=maxV(A" imB"kerE,), since only if this is so does a corrective control function u.(I) exist that, starring from -x,(O+), mainlains the plant state trajectory on Vi, hence with no influence on the error variable. By superposition, this control function keeps the error variable at zero if applied with our actual initial state (0, x.(O+), Z(o+»T, It is easily shown that u.(I) can be generated by the compensator: let us denote by ne the dimension of Vi and by T, a n, x n. basis matrix of Vi. Let F 1 be a matrix such that (A, +B,F , ) Vi C; Vi. define T = [T, T.] with T. sucb that T is nonsingular and assume as A. the first n. x n. submatrix ofT-'(A,+B,F,)T and as C. the pxn. matrix F, T ,. The initial state x.(O+) of the compensator is then given by the first n. components of - T-' x,(o+). The internal unassignable eigenvalues of Vi are the union of the elements of Z(Ap , Bp, C p) (the invariant zeros of the plant) and of the eigenvalues of the filters, that appear as invariant zeros of (A"BloE,) since filters are not controllable. Hence the corresponding eigenstructures are repeated in Ac. D
Remark. The role of the compensator is to drive the state of the plant and filters along trajectories on the (A,+B,F, )invariant Vi, thus making it possible to reproduce in the controlled plant the effect of state feedback matrix F, for every initial state belonging to Vi. Owing to Theorem 3.1 it is possible to oblain an asymptotically stable compensator if and only if the plant is minimum phase. Hence condition (14) is sufficient to ensure zero b'aCking error of the replicated reference signal with bounded input in this case. Note that F, can and must also stabilize the internal assignable eigenvalues of Vi if the controlled plant is not invertible.
1226
",
I
P"
Fig. 2. A correcting pulse with preaction. 4. USING PREACTION WHEN THE PLANT IS NON-MINIMUM PHASE The proof of Theorem 3.1 in the previous section suggests a working algorithm f(I the supervising unit to deal with exosystems state switching at t=O. Since the overall sysrem is linear and time-invatiant. multiple switchings at different instants of time are easily handled by superposition and time shifting. Note that the regulator state C
starting from -x, .• «H.) and tending to zero as time approaches infinity. while for xe•• «H.) the compensat
(a)
lIe(t)11 ~ 0
i:
oo
(b)
as tprc~-oo
11e(t)II' dt ~ 0
as t....
(25)
~ -00
(26)
Proof. Tracking error would be zero if the state of the plant (inclnding filters) at t=tprc were x,(t.... )=T,.A•.•• ".
[Xc.u~(H.)]
(27)
but, since the contrul action starts at t.... it is zero. The error transient can be computed by considering the response to -x,(tp ,,) oftbe extended plant (that, in this case. consists of plant. fillers and regulator) with tbe regulator state z(t.... ) at zero. Let us denote with v=(a" ...0;.,) the vector of the exosystems initial states and let Cl, (> 0) be the minimal real part of the eigenvalues of Ae.u; since lie .«H.) is a > 0 exists such that linear function of v. a constant
Ii, Ilx,(t....)1I :0; Ii,.····" IIvll
(23)
(28)
Let wbere A ••• has the strictly stable eigenvalues and A,.u those strictly unstable. Without loss of generality. transformation T e is assumed to be included in T ,. so that Ac has the structure shown on the right of (23) and the initial state x,«H.) of the compensator is accordingly partitioned as Xc •• «H.). x,.u«H.). Let x, •• «H.):= T,
[X,.•(~(H.)],
x'.u«H.) :=T,
[x,.u~(H.)]
Aep '= [A, +ME! B,KE, •
B,L] E t!p '[E 1 N' .-
0]
(29)
be the matrices of the extended plant The error is given by e(t-tprc )
=E.p•A.,(.-.,,,)
[-x'gpre)],
t
2:
t .... (30)
Let us denOIe by Cl, « 0) the maximal real part of the eigenvalues of A.p : a constant !i, > 0 exists such that
lIe(t)11 :0; !i, .',1'-',,,) IIx,(t....)1I :0; Ii, 50 ."('-"'.) ." ""lIvll :0; Ii, 50 ." ""lIvll
(24) As far as x, •• «H.) is concerned. it is possible to proceed as before. generating a control action u,(t) corresponding to an output-invisible state trajectory x,(t) of the plant and filters
1227
(31)
Furthermore,
[)e{tlll' dt :5 =
I.:
lIe{tlll' dt
1.'.....00 8~ ~ .'a,(I-I,,,J .'a, ""lIvll
=
S'I\'
-20,
dt
.2a, ""lIvll
(32)
The staJement directly follows from (31) and (32),
0
=
Corollary 4.1. Let V (a., .. .aq) be the vectcr of the exosystems initial states. Then, given any number e > 0, the following holds:
465.
=> lie{t)ll:5 e, t E R => where the quantities tp,.., A •• 02 . as in the proof of Theorem 4.1.
[:00 1Ie{t)II' dt :5 5•. 5, and e are
Basile, G. and G. Marro (1992). Conrrolled and Condilioned Invarianls in Unear SySlem Theory. Prentice Hall, Englewood Cliffs, New 1ersey. Davison, EJ. and B.M. Scherzinger (1987). Perfect control of the robust servomechanism jI"Oblem. IEEE Trans. Au/DmIl/. Coner., AC-31, 689-702. Devasia, S. and B. Paden (1994). Exact output traclring for oonIinear time-varying systems. Proc. IEEE Con!. on DeciJion and COnlrol, Lake Buena VISta, CA, 23462355. Devasia, S .. B. Paden and C. Rossi (1994). Minimal transient regulation of flexible suuctures. Proceedings of the IECON '94, Bologna, Italy, 1994. Francis, B.A. and W.M. Wooham (1976). The internal model principle of control theory. AU/OIIW/ica, 12,457-
£
defined
Remark. 11Ieorem 4.1 and Corollary 4.1 refer to a single switching of the exosystems leading states from zero 10 v at t - O. In the case of multiple switchings superposition applies, as specified at the beginning of this sectioo. If the preview time of the refereoce input is infinite, the em.. transienls due 10 subsequent switchings tend to reduce since the relative preaction time interval increases (with t p .. fixed); this is also true when preview time, although not infinite, is significantly greater UUI/l the time constant of the dominant right half-plane zero. However, when the preview time, hence the preaction time for each switching, is relatively short and variations of components of" are large, the error variable, while remaining bounded, may present resonance phenomena.
Gross, E., M. Tomizulra and W. Mcssner (1994). Cancellation of discrete time unstable zeros by feedforward control. ASME J. Dynamic Syst., Measuremenl Conrr., 116,33-38. Hun~ L.R., G. Meyer and R. Su (1992). Output traclring for oonIinear systems. Symposium on Implicit and Nonlinear Systems, Ariington, 'li:xas, 314-320. Isidori, A. (1985). Nonlinear control rysltm: an imroduc· tion. Springer-Verlag, Berlin. Kimwa, H. (1982). Perfect and subperfect regulation in linear multivariable control systems. AUlomalica, 18, 125-145. Marro, G. and A. Piazzi (1993). Regulation without transienlS under large parameter jumps. Proc. 12lh !FAC World Congress, 4, 23-26. Wonham, W.M. (1985). Linear Mullivariable Control: a Geometric Approach, 3rd cd. Springer-Verlag.
APPENDIX A. A MATLAB ROUTINE FOR ALGORITHM 3.1
%
• • aaxpC.(A~B,C,.,11) • • aaxpca(&',B.C .•• l). -...xl_l perfect controllability .ubapace witb r.epect to thA l-th derlvatlv. contained in ker(.).
%
Q.Xarro •
tun~t1on
5. CONCLUSION A special cootrol scheme for multivariable systems, whOSe purpose is 10 eliminate tracking error, thus also obIaining complete noointeraction, has been discussed, and some quantitative results concerning tracking error reduction in the non-minimum phase case have been presented. The application of the same technique to multivariable discretetime systemS, in which zeros outside the unit circle are quite frequent, does not require any significant change: Algorithm 3.1 Can still he applied to detect the multivariable relative degree and the corresponding relative degree condition can be satisfied by using simple delays instead of filters.
.. %
'-30-95
"_lter (II:) 1 [ .... ,DV] •• 1 •• CM) I
Bl.laa(B,O), Cl_lt.r(e), h.O, while (nv-Dw) > 0 I h •• o
DV_n.,
Z.Blt for jj.l1 Cl1-1) Z.laa([Bl,A*lnt.(lnta(Z •• ) , Cl}l.O), _nd
REFERENCES Basile, G . and G. Marro (1972). 011 the perfect output controllability of linear dynamic systems. Ricerche di AuwmDlica (in English), 2, 1-10.
A.Pl.K~l
1_.nw] •• ! •• ,W) , _Dd % --- l •• t
1228
line of
"xpc. ---