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Stabilization of multirate sampled-data linear systems
0(105-1098/90 $3.00 + 0.00 Pergamon Press pie ~) 19~1 International Federation of Automatic Control
Autoraatica, Vol. 26, No. 2. pp. 377-380, 1990 Printed in Great Britain.
Brief Paper
Stabilization of Multirate Sampled-data Linear Systems* P. COLANERI,~" R. SCATTOLINI:~ and N. SCHIAVONI§II Key Words--Sampled-data systems; multirate control systems; stabilizers; pole placement; periodic systems.
only in the past decade. For an overview of the most significant results in the area, the reader is referred to Araki and Yamamoto (1986). In that paper the authors, focusing their attention on the analysis of a control structure where a different sampling rate is associated with any pair of input-output variables, develop impulse modulation models and present criteria to assess closed-loop stability. Several,, synthesis algorithms have recently been proposed in a linear time-invariant setting for the case where all the plant outputs are sampled simultaneously at a unique constant rate, whereas the inputs are repeatedly updated between two successive plant output measurements. Among them the pole-placement approaches by Chammas and Leondes (1979) and Araki and Hagiwara (1986) have to be recalled together with the algorithms leading to LQG centralized controllers (S6derstr6m and Lennartson, 1981), generalized minimum variance self-tuners (Scattolini, 1988) and LQG decentralized regulators (Scattolini and Schiavoni, 1987; Colaneri et al., 1988). Moreover the multirate control problem has several similarities with that of improving the performance of time-invariant systems by means of periodic regulators. On this topic, see, for example, the recent papers by Khargonekar et al. (1985), Kabamba (1987) and Francis and Georgiou (1988) and the references quoted therein. In this paper the assumption is made that the plant under control is a discrete-time linear time-invariant system and a sampling mechanism is adopted permitting the various plant outputs to be measured at different rates, which can definitely be less than the unique rate adopted for the input updating, or else at different times and with a possible delay. In this framework the paper shows how a pole-placement problem can be stated, leading to a periodic controller constituted by a periodic state observer and a non-dynamic control law. The paper is organized as follows. In Section 2, the discrete-time linear time-invariant model of the plant under control is introduced and the sampling mechanism is given a precise mathematical formulation. Then, some preliminary results are presented. Finally, in Section 3 a pole-placement problem is solved by resorting to a time-varying regulator.
Abstract--This paper considers the design of multiple-input multiple-output digital control systems characterized by a non-standard sampling mechanism. It is assumed that the various outputs of the plant are measured at different rates, which can be definitively less than the unique rate a d o p t e d for the inputs updating, or else at different times. A pole-placement problem is solved by resorting to a controller composed by a periodic state observer and a non-dynamic control law. 1. Introduction
Irq SYNTHESIZING multiple-input multiple-output digital control systems, it is usually assumed that both the plant inputs updating and the plant outputs measurement are performed at a unique constant rate and in a synchronous fashion. However, this hypothesis is often not realistic or necessary for technological and/or economical reasons. These may be due to the following facts, among others: (i) Some sensors require a significant time before they supply the measurements of the plant output variables to the regulator (a typical example of such a situation arises in controlling chemical processes, where expensive chromatographs are used to measure composition products). These measurements are then infrequent and delayed with respect to those of other variables measured by sensors not suffering from such a limitation. (ii) A small number of sensors is exploited to measure a large number of output variables at different times, or the sensors allow one to measure all the plant outputs at the same rate and time, but hardware constraints prevent one from transmitting data simultaneously from all the sensors to the control processing unit. (iii) The plant outputs are all measured at the same rate and time, but this rate is less than that of the plant inputs updating allowed by the control apparatuses. It is then natural to resort to multirate control schemes, where different sampling periods are permitted for different variables along with time shifts and delays among the measurements. Research in multirate control can be traced back to the late fifties (Kranc, 1957; Kalman and Bertram, 1959; Jury, 1967); however, it has received more and more attention
2. Multirate discrete-time systems 2.1. System a n d s a m p l i n g m o d e l . Let the system under
control be described by the following discrete-time linear time-invariant model
* Received 6 June 1988; revised 15 December 1988; revised 16 May 1989; received in final form 14 June 1989. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor R. V. Patel under the direction of Editor H. Kwakernaak. ~"Centro di Teoria dei Sistemi, Consiglio Nazionale delle Ricerche, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy. :~ Dipartimento di Informatica e Sistemistica, UniversitA di Pavia, Via Abbiategrasso, 209, 27100 Pavia, Italy. § Dipartimento di Elettronica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy. II To whom correspondence should be addressed.
x ( t + 1) = A x ( t ) + B u ( t )
(la)
y(t) = C x ( t )
(lb)
where A e R n'~, B ¢ R n ' = and C ~ R "'n. Now assume that the ith component yi(t), i = 1. . . . , m , of the output vector y ( t ) is measured every T,. time instants, the T~s being positive integers. In order to account for the sampling process, we define the measured output variables wi(t), i = 1 . . . . . m , as follows w~(t):= 377
Sy,(t)
t = k T i + 3,
[o
t~kTi
+ ~i
378
Brief Paper
where k is a non-negative integer and the integers ri, 0 -< tt < T, describe the skew sampling mechanism. Then w(t) is related to the system output y(t) by w(t)
= N(t)y(t)
(2)
where
N(t):=diag {~,,(t)),
vi(t):=IX, ~
, = k T i + zi t._/:kT~+r i i = l . . . . .
m.
(3) A moment reflection shows that matrix N(t) is T-periodic, where T : = l.c.m. (T/).
Boizern and Colaneri (1987) with the same correspondence of symbols as before the result follows for t = 0. Since the detectability of any periodic pair is a time-invariant property (see, for example, Bittanti, 1986), the theorem is proved. The application of Theorem 1 calls for knowledge of the theory of linear periodic systems. However, the following sufficient conditions for the detectability of the pair (F, H) in terms of A and C can be used. Theorem 2. Suppose that (i) the pair (A, C) is detectable; (ii) there does not exist two distinct eigenvalues of A, ).i and Aj, lad-> 1, JAil-> 1, such that AS = Z~r. Then the pair (F, H) is detectable.
(4)
i=l,...,m
A quite obvious assumption on the output sampling mechanism consists of requiring that each output y~ is measured at finite intervals of time (T~ < ~). This implies that there exists at least one set of T matrices D ( i ) ~ R '~''~, i = 0 . . . . . T - 1, such that
Proof. First note that if ).i is an eigenvalue of A, then/~i = AS is an ¢igenvalu¢ of F = A r. Assume now that A is in the Jordan canonical form A : = d i a g (Ai},
i = 1. . . . .
h,
where ~i is the number of distinct eigenvalues. Then
T-I
~ i D(i)N(i) = L
(5)
A i : = d i a g (Aq},
] = 1. . . . .
h/,
0
By combining the model (1) and the sampling mechanism (2)-(4), the system to be controlled turns out to be T-periodic in the output transformation. Observe that the possible presence of time delays in the output measures has not been taken into explicit account in the considered model for the sake of simplicity, but it can be dealt with by simply augmenting the dimension of the model state space (Scattolini and Schiavoni, 1987). 2.2. Time-invariant reformulation and structural properties. A common attitude in anr..lysing linear periodic systems is to resort to the time-invadant reformulation first introduoed in Meyer and Burrus (1975) (see also Khargonekar et al., 1985). Along those lines, let k be a new discrete-time variable and define
where hi is the geometric multiplicity of the eigenvalue ).i and -A i
.. •
0-
0
~'i I
1
0
"."
0
0
0
."
Zi
Ao:=
0
It trivially follows that A r=diag(AT},
i=l .....
fi,
~l(k):=x(kT) O(k):= lu'(kT) u ' ( k T + 1 ) . . . u ' ( k V + T - 1)1' y(k):=lw'(kT) w'(kT + 1)... w'(kT + T-
0
1)l'.
Then a "T-sampled" representation of system (1)-(4) is given by r/(k + 1) = Frl(k) + GO(k)
(6a)
y(k) = H~l(k) + EO(k)
(6b)
where
(6c)
F:=A r G:=IGo G t ' "
Gr-tl,
Gi:=Ar-i-lB,
i=0 .....
T-1
(~) H:ffi IHUV'(0) HiN'(1) • • • H ~ . _ t N ' ( T - 1)l', H i : = ¢ A ~, i = 0 . . . . .
T-1
(6¢)
and E : = (Eq},
_ fN(i-1)CAi-i-~B /:'# := ~0 i=1 .....
i>j i<] T,
j=l .....
T.
AT
*
""
*
o
o
...
AT~
A~= -o
If Ai ~ 0 the elements denoted by * are easily proven to be non-zero and there exists a vector tpq ~ 0 such that Aijq~q = ~u~il i f f A ~ % = 2T%r In view of (ii), the number and dimensions of the blocks of A r associated with AT, IAA> 1, coincide with the ones associated with ).i. Then, in view of the block-diagonal structure of both A and A r, it can be concluded that the set of the h i eigenvectors of A corresponding to ),i coincides with the set of the ~ eigenvectors of A r corresponding to 2~r. Assume now by contradiction that ( F , H ) is not detectable. Then, following the well known PBH test (see, for example, Kailath, 1980), for some /~ and tp, I/al ~ 1 and q~~ 0, it turns out that
(60
AT~ = [~
N(i)CAiq~=O
i=O .....
T-1.
In the next section, the stabilization problem for system (1)-(4) will be solved by resorting to reformulation (6). Hence, the relationship between the observability property of system (6) and the corresponding one of the underlying T-periodic model (1)-(4) is now investigated.
From the discussion above it follows that
Theorem 1. (i) The unobservability subspace of the pair (F, H) coincides with the unobservability subspaee of the pair (A, N(.)C) at t = 0. (ii) The pair (F, H) is detectable iff the pair (A, N(.)C) is detectable.
where A =/z t/T, JAI-> 1. From N(i)Ccp = 0 and by recalling the definition (5) of the D(i)s, one obtains
Proof. (i) The proof follows from Lemma 4 in Bolzern and Colaneri (1987), by noting that the symbols ~o and/~o there correspond to F and H here. (ii) From Lemma 3, Part (iii) in
Atp = Acp N(i)CAicp=O
i=O .....
T-1
T-I
~ , D(i)N(i)C~p = Cq9 = 0 0
which leads to contradiction.
Brief Paper It is worth noting that the detectability condition above is largely independent of the sampling mechanism, in that it does not involve the integers T~and ~, but only the period T defined by (4).
3. Pole.placement via output feedback The aim of this section is to determine a regulator for system (1)-(4) which assigns the eigenvalues of the monodromy matrix associated with the closed-loop system. Recall that, given a discrete-time T-periodic matrix A(.), the monodromy matrix associated with A(-) is defined by OA(T, 0 ) : = A ( T - 1)A(T - 2 ) . . . A(0). As is well known, matrix A(-), or, equivalently, the system associated with it, is asymptotically stable iff the eigenvalues of ~A(T, 0) (characteristic multiplers) lie inside the open unit disk of the complex plane. 3.1. State observer. Consider the following observer for the time invariant system (6): /'/(k + 1) = FO(k) + GO(k) + £(HO(k ) + EO(k) - y(k)), (7) where/.~ ~ R n'mr is an arbitrarily chosen matrix, which can be partitioned as £:=l£o£t..-£r_d,
£ , e R n'",
i=0 .....
T-1.
(8)
System (7) is characterized by the dynamic matrix F + £H, which, by recalling (6c) and (6¢), can be written as T-I
F + £ H = A r + ~ , £,N(i)H,.
(9)
0
In order to relate (7) with the T-periodic system (1)-(4), let L(.) be the T-periodic matrix function such that L(i):= £,N(i),
i = 0. . . . .
r - 1.
(10)
Moreover, define two T-periodic n x n matrix functions S(.) and Q(.) such that
l-Q(i):=S(i):={~;
i=O i=l .....
(I1)
T-1
379
seems to be the price that has to be paid in order to assign the poles of a periodic observer. Indeed, a somehow similar approach has been used by Kabamba (1986) in solving a (dual) pole-placement problem by state feedback for periodic systems. An alternative to the deterministic approach used here consists of resorting to the optimal stochastic periodic prediction theory (Amit and Powell, 1981; Bittanti et al., 1988). Of course, proceeding this way, the positions of the predictor poles are not arbitrarily assigned. Let n - n o be the dimension of the unobservability subspace of (F, H). Then, in view of Theorems 1 and 3, it turns out that the characteristic multipliers of At(') can be partitioned as follows: (i) n equal to zero; (ii) n - n o coinciding with the eigenvalues of the unobservable part of the pair (A, N(.)C); (iii) n o arbitrarily placed (in complex conjugate pairs) by a suitable choice of L(.). In particular, for stabilization, the following result holds.
Corollary 1. There exists a T-periodic matrix L(.) such the characteristic multipliers of A~(.) lie inside the unit iff the pair (A, N(.)C) is detectable. 3.2. State feedback. Let n, be the dimension of teachability subspace of the pair (A, B). Then, it is very known that there exists a control law u(t) = Kx(t),
K ~ R"",
that disk the well (14)
such that the closed-loop system (la), (14) possesses: (i) n, eigenvalues arbitrarily placed (in complex conjugate pairs); (ii) n - n , eigenvalues coinciding with those of the unreachable part of (A, B). 3.3. Output feedback. Consider the T-periodic regulator constructed from (10)-(12) and (14) .~(t + 1) = A.~(t) + Bu(t) + Q(t + 1)r(t) + Q(t + 1)L(t)[C.~(t) - w(t)]
(15a)
r(t + 1) = S(t)r(t) + L(t)[C.~(t) - w(/)]
(15b)
u(t) = K(t).~(t).
(15c)
Now, referring the closed-loop system (1)-(4), (15) to the state vector [~'r'x']', the corresponding T-periodic state matrix takes on the form
and consider the T-periodic system .¢(t + 1) = A~(t) + Bu(t) + Q(t + 1)r(t + 1)
(12a)
r(t + 1) = S(t)r(t) + L(t)[C~(t) - w(t)].
(12b)
A2(t):=[At~ t) A ? B K ] " .,
Since Q(t + 1)S(t)= Q(t + 1), system (12) turns out to be characterized by the T-periodic state matrix
At(t):=[A+Q(t+I)L(t)CQ(t + 1)] L(t)C
S(t)
(13)
J"
Theorem 3. The matrix A~(.) possesses n characteristic multipliers coinciding with the n eigenvalues of matrix F +/~H and n characteristic multipliers equal to zero. Proof. The monodromy matrix ~A~(T, 0) of At(. ) is given by ~,,~(T, O) =
A t + ~,~ £,N(i)Hi o ?
•
Hence, (9) leads to the conclusion.
Remark 1. Notice that, letting ~ : = $ - x and q:=[¢'r']', it results that q(t + 1)= A~(t)q(t). Hence, if At(t ) is asymptotically stable, it follows that the reconstruction error ~ asymptotically vanishes. Moreover, e((k + 1)T)= (F + £H)~(kT), so that .~(kT) = O(k). Remark 2. Notice that the structure of system (12) is such that ~c(kT + i), i = 0 . . . . . T - 1, actually depends on the measurements w(t) up to the time t = k T - 1. This fact is not completely satisfactory from a practical point of view, but AUTO 26:~*N
Then the result below easily follows from Theorem 2.
Theorem 4. The matrix A2(. ) possesses n characteristic multipliers coinciding with the n eigenvalues of matrix ( A + B K ) r, n characteristic multipliers coinciding with the n eigenvalues of matrix (F + LH) and n characteristic multipliers equal to zero. Notice that by suitably selecting matrix K and the T-periodic matrix L(.), it is possible to arbitrarily assign n , + n o characteristic multipliers of A2(. ). Moreover, n characteristic multipliers of A2(- ) are equal to zero while 2n - n, - n o belong to the unreachable part of (A, B) and/or to the unobservable part at t = 0 of (A, N(.)C). As a conclusion, it is possible to state the main result of this section. Theorem 5. There exists a matrix K and a T-periodic matrix L(.) such that the closed-loop system (1)-(4), (15) is asymptotically stable iff the pair (A, B) is stabilizable and the pair (A, N(.)C) is detectable. 4. Conclusions In this paper a multirate pole-placement control algorithm for discrete-time linear multivariable systems has been presented. The assumption is made that the inputs are allowed to be modified at any sampling period, while the outputs are measured at longer intervals and, possibly, not simultaneously. Many other streams of research can be considered concerning multirate control. Among them, the following
380
Brief Paper
ones are worth mention!rig: (1) the analysis of more complex cases where constraints, typically due to the actuators, have to be imposed both on the frequency of the updating and the value of the control variables; (2) the development of multirate controllers based on adaptive control techniques to handle the cases where information on the process is lacking. Acknowledgements--This paper has been partially supported by MP! and Centro di Teoria dei Sistemi, CNR. References Amit, N. and J. D. Powell (1981). Optimal digital control of multirate systems. Proc. A I A A Guidance and Control Conf., Albuquerque, pp. 423-429. Araki, M. and T. Hagiwara (1986). Pole assignment by multirate sampled-data output feedback. Int. J. Control, 44, 1661-1673. Araki, M. and K. Yamamoto (1986). Multivariable multirate sampled-data systems: state-space description, transfer characteristics, and Nyquist criterion. I E E E Trans. Aut. Control, AC-31, 145-154. Bittanti, S. (1986). Deterministic and stochastic linear periodic systems. In Time Series and Linear Systems, pp. 141-182. Springer-Verlag. Bittanti, S., P. Colaneri and G. De Nicolao (1988). The difference periodic Riccati equation for the periodic prediction probleln. IEEE Trans. Aut. Control, AC-33, 706-712. Boizern, P. and P. Colaneri (1987). Inertia theorems for the periodic Lyapunov difference equation and periodic Riccati difference equation. Linear Algebra and its Applications, 85, 249-265. Chammas, A. B. and C. Leondes (1979). Pole assignment by piecewise constant output feedback. Int. J. Control, 29, 31-38. Colaneri, P., R. Scattolini and N. Schiavoni (1988). A design technique for multirate control with application to a
distillation column. Proc. 12th IMACS World Congr., Vol. 2, pp. 589-591. Francis, B. A. and T. T. Georgiou (1988). Stability theory for linear time-invariant plants with periodic digital controllers. IEEE Trans. Aut. Control, AC-33, 820-832. Jury, E. I. (1967). A note on multirate sampled-data systems. IEEE Trans. Aut. Control, AC-12, 319-320. Kabamba, P. T. (1986). Monodromy eigenvalue assignment in linear periodic systems. IEEE Trans. Aut. Control, AC-31, 950-952. Kabamba, P. T. (1987). Control of linear systems using generalized sampled-data hold functions. IEEE Trans. Aut. Control, AC-32, 772-783. Kailath, T. (1980). Linear Systems. Prentice-Hall, Englewood Cliffs, NJ. Kalman, P,. E. and J. E. Bertram (1959). A unified approach to the theory of sampling systems. J. Franklin Inst., 267, 405-436. Khargonekar, P. P., K. Poola and A. Tannenbaum (1985). Robust control of linear time-invariant plants using periodic compensation. IEEE Trans. Aut. Control, AC-30, 1088-1096. Kranc, G. M. (1957). Input-output analysis of multirate feedback system. IRE Trans. Aut. Control, 3, 21-28. Meyer, R. A. and C. S. Burrus (1975). A unified analysis of multirate and periodically time-varying digital filters. IEEE Trans. Ccts Syst., CAS-22, 162-168. Scattolini, R. (1988). Self-tuning control of systems with infrequent and delayed output sampling. Proc. lEE-D, 135, 213-221. Scattolini, R. and N. Schiavoni (1987). Design of multirate control systems via parameter optimization. 26th IEEE CDC, Vol. 2, pp. 1556-1557. S6derstr6m, T. and B. Lennartson (1981). On linear optimal control with infrequent output sampling. Proc. 3rd IMA Conf. on Control Theory, pp. 605-624. Academic Press, New York.
Autoraatica, Vol. 26, No. 2. pp. 377-380, 1990 Printed in Great Britain.
Brief Paper
Stabilization of Multirate Sampled-data Linear Systems* P. COLANERI,~" R. SCATTOLINI:~ and N. SCHIAVONI§II Key Words--Sampled-data systems; multirate control systems; stabilizers; pole placement; periodic systems.
only in the past decade. For an overview of the most significant results in the area, the reader is referred to Araki and Yamamoto (1986). In that paper the authors, focusing their attention on the analysis of a control structure where a different sampling rate is associated with any pair of input-output variables, develop impulse modulation models and present criteria to assess closed-loop stability. Several,, synthesis algorithms have recently been proposed in a linear time-invariant setting for the case where all the plant outputs are sampled simultaneously at a unique constant rate, whereas the inputs are repeatedly updated between two successive plant output measurements. Among them the pole-placement approaches by Chammas and Leondes (1979) and Araki and Hagiwara (1986) have to be recalled together with the algorithms leading to LQG centralized controllers (S6derstr6m and Lennartson, 1981), generalized minimum variance self-tuners (Scattolini, 1988) and LQG decentralized regulators (Scattolini and Schiavoni, 1987; Colaneri et al., 1988). Moreover the multirate control problem has several similarities with that of improving the performance of time-invariant systems by means of periodic regulators. On this topic, see, for example, the recent papers by Khargonekar et al. (1985), Kabamba (1987) and Francis and Georgiou (1988) and the references quoted therein. In this paper the assumption is made that the plant under control is a discrete-time linear time-invariant system and a sampling mechanism is adopted permitting the various plant outputs to be measured at different rates, which can definitely be less than the unique rate adopted for the input updating, or else at different times and with a possible delay. In this framework the paper shows how a pole-placement problem can be stated, leading to a periodic controller constituted by a periodic state observer and a non-dynamic control law. The paper is organized as follows. In Section 2, the discrete-time linear time-invariant model of the plant under control is introduced and the sampling mechanism is given a precise mathematical formulation. Then, some preliminary results are presented. Finally, in Section 3 a pole-placement problem is solved by resorting to a time-varying regulator.
Abstract--This paper considers the design of multiple-input multiple-output digital control systems characterized by a non-standard sampling mechanism. It is assumed that the various outputs of the plant are measured at different rates, which can be definitively less than the unique rate a d o p t e d for the inputs updating, or else at different times. A pole-placement problem is solved by resorting to a controller composed by a periodic state observer and a non-dynamic control law. 1. Introduction
Irq SYNTHESIZING multiple-input multiple-output digital control systems, it is usually assumed that both the plant inputs updating and the plant outputs measurement are performed at a unique constant rate and in a synchronous fashion. However, this hypothesis is often not realistic or necessary for technological and/or economical reasons. These may be due to the following facts, among others: (i) Some sensors require a significant time before they supply the measurements of the plant output variables to the regulator (a typical example of such a situation arises in controlling chemical processes, where expensive chromatographs are used to measure composition products). These measurements are then infrequent and delayed with respect to those of other variables measured by sensors not suffering from such a limitation. (ii) A small number of sensors is exploited to measure a large number of output variables at different times, or the sensors allow one to measure all the plant outputs at the same rate and time, but hardware constraints prevent one from transmitting data simultaneously from all the sensors to the control processing unit. (iii) The plant outputs are all measured at the same rate and time, but this rate is less than that of the plant inputs updating allowed by the control apparatuses. It is then natural to resort to multirate control schemes, where different sampling periods are permitted for different variables along with time shifts and delays among the measurements. Research in multirate control can be traced back to the late fifties (Kranc, 1957; Kalman and Bertram, 1959; Jury, 1967); however, it has received more and more attention
2. Multirate discrete-time systems 2.1. System a n d s a m p l i n g m o d e l . Let the system under
control be described by the following discrete-time linear time-invariant model
* Received 6 June 1988; revised 15 December 1988; revised 16 May 1989; received in final form 14 June 1989. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor R. V. Patel under the direction of Editor H. Kwakernaak. ~"Centro di Teoria dei Sistemi, Consiglio Nazionale delle Ricerche, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy. :~ Dipartimento di Informatica e Sistemistica, UniversitA di Pavia, Via Abbiategrasso, 209, 27100 Pavia, Italy. § Dipartimento di Elettronica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy. II To whom correspondence should be addressed.
x ( t + 1) = A x ( t ) + B u ( t )
(la)
y(t) = C x ( t )
(lb)
where A e R n'~, B ¢ R n ' = and C ~ R "'n. Now assume that the ith component yi(t), i = 1. . . . , m , of the output vector y ( t ) is measured every T,. time instants, the T~s being positive integers. In order to account for the sampling process, we define the measured output variables wi(t), i = 1 . . . . . m , as follows w~(t):= 377
Sy,(t)
t = k T i + 3,
[o
t~kTi
+ ~i
378
Brief Paper
where k is a non-negative integer and the integers ri, 0 -< tt < T, describe the skew sampling mechanism. Then w(t) is related to the system output y(t) by w(t)
= N(t)y(t)
(2)
where
N(t):=diag {~,,(t)),
vi(t):=IX, ~
, = k T i + zi t._/:kT~+r i i = l . . . . .
m.
(3) A moment reflection shows that matrix N(t) is T-periodic, where T : = l.c.m. (T/).
Boizern and Colaneri (1987) with the same correspondence of symbols as before the result follows for t = 0. Since the detectability of any periodic pair is a time-invariant property (see, for example, Bittanti, 1986), the theorem is proved. The application of Theorem 1 calls for knowledge of the theory of linear periodic systems. However, the following sufficient conditions for the detectability of the pair (F, H) in terms of A and C can be used. Theorem 2. Suppose that (i) the pair (A, C) is detectable; (ii) there does not exist two distinct eigenvalues of A, ).i and Aj, lad-> 1, JAil-> 1, such that AS = Z~r. Then the pair (F, H) is detectable.
(4)
i=l,...,m
A quite obvious assumption on the output sampling mechanism consists of requiring that each output y~ is measured at finite intervals of time (T~ < ~). This implies that there exists at least one set of T matrices D ( i ) ~ R '~''~, i = 0 . . . . . T - 1, such that
Proof. First note that if ).i is an eigenvalue of A, then/~i = AS is an ¢igenvalu¢ of F = A r. Assume now that A is in the Jordan canonical form A : = d i a g (Ai},
i = 1. . . . .
h,
where ~i is the number of distinct eigenvalues. Then
T-I
~ i D(i)N(i) = L
(5)
A i : = d i a g (Aq},
] = 1. . . . .
h/,
0
By combining the model (1) and the sampling mechanism (2)-(4), the system to be controlled turns out to be T-periodic in the output transformation. Observe that the possible presence of time delays in the output measures has not been taken into explicit account in the considered model for the sake of simplicity, but it can be dealt with by simply augmenting the dimension of the model state space (Scattolini and Schiavoni, 1987). 2.2. Time-invariant reformulation and structural properties. A common attitude in anr..lysing linear periodic systems is to resort to the time-invadant reformulation first introduoed in Meyer and Burrus (1975) (see also Khargonekar et al., 1985). Along those lines, let k be a new discrete-time variable and define
where hi is the geometric multiplicity of the eigenvalue ).i and -A i
.. •
0-
0
~'i I
1
0
"."
0
0
0
."
Zi
Ao:=
0
It trivially follows that A r=diag(AT},
i=l .....
fi,
~l(k):=x(kT) O(k):= lu'(kT) u ' ( k T + 1 ) . . . u ' ( k V + T - 1)1' y(k):=lw'(kT) w'(kT + 1)... w'(kT + T-
0
1)l'.
Then a "T-sampled" representation of system (1)-(4) is given by r/(k + 1) = Frl(k) + GO(k)
(6a)
y(k) = H~l(k) + EO(k)
(6b)
where
(6c)
F:=A r G:=IGo G t ' "
Gr-tl,
Gi:=Ar-i-lB,
i=0 .....
T-1
(~) H:ffi IHUV'(0) HiN'(1) • • • H ~ . _ t N ' ( T - 1)l', H i : = ¢ A ~, i = 0 . . . . .
T-1
(6¢)
and E : = (Eq},
_ fN(i-1)CAi-i-~B /:'# := ~0 i=1 .....
i>j i<] T,
j=l .....
T.
AT
*
""
*
o
o
...
AT~
A~= -o
If Ai ~ 0 the elements denoted by * are easily proven to be non-zero and there exists a vector tpq ~ 0 such that Aijq~q = ~u~il i f f A ~ % = 2T%r In view of (ii), the number and dimensions of the blocks of A r associated with AT, IAA> 1, coincide with the ones associated with ).i. Then, in view of the block-diagonal structure of both A and A r, it can be concluded that the set of the h i eigenvectors of A corresponding to ),i coincides with the set of the ~ eigenvectors of A r corresponding to 2~r. Assume now by contradiction that ( F , H ) is not detectable. Then, following the well known PBH test (see, for example, Kailath, 1980), for some /~ and tp, I/al ~ 1 and q~~ 0, it turns out that
(60
AT~ = [~
N(i)CAiq~=O
i=O .....
T-1.
In the next section, the stabilization problem for system (1)-(4) will be solved by resorting to reformulation (6). Hence, the relationship between the observability property of system (6) and the corresponding one of the underlying T-periodic model (1)-(4) is now investigated.
From the discussion above it follows that
Theorem 1. (i) The unobservability subspace of the pair (F, H) coincides with the unobservability subspaee of the pair (A, N(.)C) at t = 0. (ii) The pair (F, H) is detectable iff the pair (A, N(.)C) is detectable.
where A =/z t/T, JAI-> 1. From N(i)Ccp = 0 and by recalling the definition (5) of the D(i)s, one obtains
Proof. (i) The proof follows from Lemma 4 in Bolzern and Colaneri (1987), by noting that the symbols ~o and/~o there correspond to F and H here. (ii) From Lemma 3, Part (iii) in
Atp = Acp N(i)CAicp=O
i=O .....
T-1
T-I
~ , D(i)N(i)C~p = Cq9 = 0 0
which leads to contradiction.
Brief Paper It is worth noting that the detectability condition above is largely independent of the sampling mechanism, in that it does not involve the integers T~and ~, but only the period T defined by (4).
3. Pole.placement via output feedback The aim of this section is to determine a regulator for system (1)-(4) which assigns the eigenvalues of the monodromy matrix associated with the closed-loop system. Recall that, given a discrete-time T-periodic matrix A(.), the monodromy matrix associated with A(-) is defined by OA(T, 0 ) : = A ( T - 1)A(T - 2 ) . . . A(0). As is well known, matrix A(-), or, equivalently, the system associated with it, is asymptotically stable iff the eigenvalues of ~A(T, 0) (characteristic multiplers) lie inside the open unit disk of the complex plane. 3.1. State observer. Consider the following observer for the time invariant system (6): /'/(k + 1) = FO(k) + GO(k) + £(HO(k ) + EO(k) - y(k)), (7) where/.~ ~ R n'mr is an arbitrarily chosen matrix, which can be partitioned as £:=l£o£t..-£r_d,
£ , e R n'",
i=0 .....
T-1.
(8)
System (7) is characterized by the dynamic matrix F + £H, which, by recalling (6c) and (6¢), can be written as T-I
F + £ H = A r + ~ , £,N(i)H,.
(9)
0
In order to relate (7) with the T-periodic system (1)-(4), let L(.) be the T-periodic matrix function such that L(i):= £,N(i),
i = 0. . . . .
r - 1.
(10)
Moreover, define two T-periodic n x n matrix functions S(.) and Q(.) such that
l-Q(i):=S(i):={~;
i=O i=l .....
(I1)
T-1
379
seems to be the price that has to be paid in order to assign the poles of a periodic observer. Indeed, a somehow similar approach has been used by Kabamba (1986) in solving a (dual) pole-placement problem by state feedback for periodic systems. An alternative to the deterministic approach used here consists of resorting to the optimal stochastic periodic prediction theory (Amit and Powell, 1981; Bittanti et al., 1988). Of course, proceeding this way, the positions of the predictor poles are not arbitrarily assigned. Let n - n o be the dimension of the unobservability subspace of (F, H). Then, in view of Theorems 1 and 3, it turns out that the characteristic multipliers of At(') can be partitioned as follows: (i) n equal to zero; (ii) n - n o coinciding with the eigenvalues of the unobservable part of the pair (A, N(.)C); (iii) n o arbitrarily placed (in complex conjugate pairs) by a suitable choice of L(.). In particular, for stabilization, the following result holds.
Corollary 1. There exists a T-periodic matrix L(.) such the characteristic multipliers of A~(.) lie inside the unit iff the pair (A, N(.)C) is detectable. 3.2. State feedback. Let n, be the dimension of teachability subspace of the pair (A, B). Then, it is very known that there exists a control law u(t) = Kx(t),
K ~ R"",
that disk the well (14)
such that the closed-loop system (la), (14) possesses: (i) n, eigenvalues arbitrarily placed (in complex conjugate pairs); (ii) n - n , eigenvalues coinciding with those of the unreachable part of (A, B). 3.3. Output feedback. Consider the T-periodic regulator constructed from (10)-(12) and (14) .~(t + 1) = A.~(t) + Bu(t) + Q(t + 1)r(t) + Q(t + 1)L(t)[C.~(t) - w(t)]
(15a)
r(t + 1) = S(t)r(t) + L(t)[C.~(t) - w(/)]
(15b)
u(t) = K(t).~(t).
(15c)
Now, referring the closed-loop system (1)-(4), (15) to the state vector [~'r'x']', the corresponding T-periodic state matrix takes on the form
and consider the T-periodic system .¢(t + 1) = A~(t) + Bu(t) + Q(t + 1)r(t + 1)
(12a)
r(t + 1) = S(t)r(t) + L(t)[C~(t) - w(t)].
(12b)
A2(t):=[At~ t) A ? B K ] " .,
Since Q(t + 1)S(t)= Q(t + 1), system (12) turns out to be characterized by the T-periodic state matrix
At(t):=[A+Q(t+I)L(t)CQ(t + 1)] L(t)C
S(t)
(13)
J"
Theorem 3. The matrix A~(.) possesses n characteristic multipliers coinciding with the n eigenvalues of matrix F +/~H and n characteristic multipliers equal to zero. Proof. The monodromy matrix ~A~(T, 0) of At(. ) is given by ~,,~(T, O) =
A t + ~,~ £,N(i)Hi o ?
•
Hence, (9) leads to the conclusion.
Remark 1. Notice that, letting ~ : = $ - x and q:=[¢'r']', it results that q(t + 1)= A~(t)q(t). Hence, if At(t ) is asymptotically stable, it follows that the reconstruction error ~ asymptotically vanishes. Moreover, e((k + 1)T)= (F + £H)~(kT), so that .~(kT) = O(k). Remark 2. Notice that the structure of system (12) is such that ~c(kT + i), i = 0 . . . . . T - 1, actually depends on the measurements w(t) up to the time t = k T - 1. This fact is not completely satisfactory from a practical point of view, but AUTO 26:~*N
Then the result below easily follows from Theorem 2.
Theorem 4. The matrix A2(. ) possesses n characteristic multipliers coinciding with the n eigenvalues of matrix ( A + B K ) r, n characteristic multipliers coinciding with the n eigenvalues of matrix (F + LH) and n characteristic multipliers equal to zero. Notice that by suitably selecting matrix K and the T-periodic matrix L(.), it is possible to arbitrarily assign n , + n o characteristic multipliers of A2(. ). Moreover, n characteristic multipliers of A2(- ) are equal to zero while 2n - n, - n o belong to the unreachable part of (A, B) and/or to the unobservable part at t = 0 of (A, N(.)C). As a conclusion, it is possible to state the main result of this section. Theorem 5. There exists a matrix K and a T-periodic matrix L(.) such that the closed-loop system (1)-(4), (15) is asymptotically stable iff the pair (A, B) is stabilizable and the pair (A, N(.)C) is detectable. 4. Conclusions In this paper a multirate pole-placement control algorithm for discrete-time linear multivariable systems has been presented. The assumption is made that the inputs are allowed to be modified at any sampling period, while the outputs are measured at longer intervals and, possibly, not simultaneously. Many other streams of research can be considered concerning multirate control. Among them, the following
380
Brief Paper
ones are worth mention!rig: (1) the analysis of more complex cases where constraints, typically due to the actuators, have to be imposed both on the frequency of the updating and the value of the control variables; (2) the development of multirate controllers based on adaptive control techniques to handle the cases where information on the process is lacking. Acknowledgements--This paper has been partially supported by MP! and Centro di Teoria dei Sistemi, CNR. References Amit, N. and J. D. Powell (1981). Optimal digital control of multirate systems. Proc. A I A A Guidance and Control Conf., Albuquerque, pp. 423-429. Araki, M. and T. Hagiwara (1986). Pole assignment by multirate sampled-data output feedback. Int. J. Control, 44, 1661-1673. Araki, M. and K. Yamamoto (1986). Multivariable multirate sampled-data systems: state-space description, transfer characteristics, and Nyquist criterion. I E E E Trans. Aut. Control, AC-31, 145-154. Bittanti, S. (1986). Deterministic and stochastic linear periodic systems. In Time Series and Linear Systems, pp. 141-182. Springer-Verlag. Bittanti, S., P. Colaneri and G. De Nicolao (1988). The difference periodic Riccati equation for the periodic prediction probleln. IEEE Trans. Aut. Control, AC-33, 706-712. Boizern, P. and P. Colaneri (1987). Inertia theorems for the periodic Lyapunov difference equation and periodic Riccati difference equation. Linear Algebra and its Applications, 85, 249-265. Chammas, A. B. and C. Leondes (1979). Pole assignment by piecewise constant output feedback. Int. J. Control, 29, 31-38. Colaneri, P., R. Scattolini and N. Schiavoni (1988). A design technique for multirate control with application to a
distillation column. Proc. 12th IMACS World Congr., Vol. 2, pp. 589-591. Francis, B. A. and T. T. Georgiou (1988). Stability theory for linear time-invariant plants with periodic digital controllers. IEEE Trans. Aut. Control, AC-33, 820-832. Jury, E. I. (1967). A note on multirate sampled-data systems. IEEE Trans. Aut. Control, AC-12, 319-320. Kabamba, P. T. (1986). Monodromy eigenvalue assignment in linear periodic systems. IEEE Trans. Aut. Control, AC-31, 950-952. Kabamba, P. T. (1987). Control of linear systems using generalized sampled-data hold functions. IEEE Trans. Aut. Control, AC-32, 772-783. Kailath, T. (1980). Linear Systems. Prentice-Hall, Englewood Cliffs, NJ. Kalman, P,. E. and J. E. Bertram (1959). A unified approach to the theory of sampling systems. J. Franklin Inst., 267, 405-436. Khargonekar, P. P., K. Poola and A. Tannenbaum (1985). Robust control of linear time-invariant plants using periodic compensation. IEEE Trans. Aut. Control, AC-30, 1088-1096. Kranc, G. M. (1957). Input-output analysis of multirate feedback system. IRE Trans. Aut. Control, 3, 21-28. Meyer, R. A. and C. S. Burrus (1975). A unified analysis of multirate and periodically time-varying digital filters. IEEE Trans. Ccts Syst., CAS-22, 162-168. Scattolini, R. (1988). Self-tuning control of systems with infrequent and delayed output sampling. Proc. lEE-D, 135, 213-221. Scattolini, R. and N. Schiavoni (1987). Design of multirate control systems via parameter optimization. 26th IEEE CDC, Vol. 2, pp. 1556-1557. S6derstr6m, T. and B. Lennartson (1981). On linear optimal control with infrequent output sampling. Proc. 3rd IMA Conf. on Control Theory, pp. 605-624. Academic Press, New York.