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Automotica, Vol. 31, No. II, pp. 160-1617, 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved cn?o5-1098/95 $9.50 + 0.00
Efficient Eigenvalue Assignments for General Linear MIMO Systems* MICHAEL
VAL&EKt$
and NEJAT
OLGACt
A general class of time-varying linear MZMO systems is considered. These are first tranformed into Frobenius form. Then a feedback pole placement technique is implemented in the sense of Lyapunov equivalence. A step-by-step development is given for SZSO time-invariant and timevarying and for MZMO time-invariant and time-varying systems. Key Words-Pole
assignment; eigenvalue placement; Lyapunov equivalent; Frobenius form; linear MIMO system: feedback stabilization; characteristic equation; linear SISO systems.
Abstrad-This paper deals with the transformation of linear, multi-input multi-output (MIMO) systems into Frobenius canonical form, with the ultimate objective of developing a new, computationally efficient methodology for a poleplacement procedure. Both time-invariant and time-varying systems are considered. The conventional pole placement steps for time-invariant SISO (single-input single-output) systems are generalized for both classes. This is a unique study of the expansion of the pole placement capability, in particular for time-varying MIMO systems. This depth of generalization has been neglected in the past due to its complex formulation. The practical advantage of the proposed technique is that it does not require the coefficients of the characteristic polynomial, the eigenvalues of the original system, or the coefficients of the characteristic polynomial of the transformed system. The repercussions of such a development are expected in nonlinear systems theory as well, considering the fact that some recently developed techniques yield exact I/O linearization with time-varying coefficients.
(in particular, Frobenius and Hessenberg) have been treated extensively in the literature (Silverman, 1966; Tuel, 1966; Luenberger, 1967; Kailath, 1980; Varga, 1981; Kuo, 1982; Miminis and Paige, 1982; Petkov et al., 1984, 1985; Kautsky et al., 1985). In this study, particular attention is directed towards the Frobenius canonical form, because of its unparalleled position in arriving at the desired pole placement for time-varying systems. With awareness of the numerical instabilities arising during this procedure (particularly for higher-dimensional states), we take the necessary steps towards developing a technique to handle linear time-varying MIMO systems. For the curious reader, a treatment that alleviates this instability for SISO structures can be found in ValaSek and Olgac (1995). It should also be noted that at present there is no transformation that is numerically robust for linear time-varying systems. Generally pole placement requires the computation of the characteristic polynomial coefficients for either the original or the new state matrices or the eigenvalues of the original system matrix. Blanchini (1989) introduced a technique that removes this requirement for SISO (single-input single-output) systems by utilizing an intermediate transformation to a bidiagonal Frobenius form. This technique has been further extended for time-varying linear systems in Tsai et al., 1991. Nguyen (1987) introduced the Frobenius transformation for time-varying systems; however, his treatment requires that the characteristic polynomial coefficients of the desired behavior be computed as well as the complete Frobenius transformation of the system. All these requirements are
1. INTRODUCTION
This paper deals primarily with the transformation of general linear multi-input multi-output (MIMO) systems into Frobenius canonical form. In this transformed domain, a new, computationally efficient, state feedback structure is introduced for a desired output behavior. Similar transformations into different canonical forms *Received 15 April 1993; revised 14 September 1994; received in final form 7 June 1995. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor V. Ku&a under the direction of Editor Huibert Kwakernaak. Corresponding author Professor Nejat Olgac. Tel. + 1203 486 2382; Fax + 1203 486 5088; E-mail
[email protected]. t University of Connecticut, Mechanical Engineering Department, Storrs, CT 06269-3139, U.S.A. $ Permanent address: Faculty of Mechanical Engineering, Department of Mechanics, Czech Technical University, Prague, Czech Republic. 1605
1606
M. Val&Zek and N. Olgac
removed in the technique presented in this paper. Investigations of linear MIMO time-varying systems by Nguyen (1987) resulted in a claim that his methodology does not function except for some particular class of controllability matrix structures. The steps of Tsai et al. (1991) require that the transformation of the system dynamics into the bidiagonal Frobenius form be done through a time-invariant transformation matrix. Both of these conditions are very restrictive, and can easily be violated. In contrast, the procedure detailed in this paper has a very general domain of validity, so long as the combined system considered is controllable. Therefore the procedure defined here represents a unique treatment for MIMO systems in the literature. It is clear that both the transformation and pole-placement efforts are of significant and continuing interest to the systems and controls community. Our current work brings a different perspective to addressing these issues and, furthermore, it offers a unique and numerically simplified procedure to achieve this objective. In summary, the text consists of the following. In Section 2 a brief summary of results for the SISO time-invariant and time-varying case is presented. Section 3 deals with the basics of the time-invariant MIMO pole-placement procedures, with the following highlights:
G-4the
procedure presented here provides a simple generalization of classical Ackermann’s formula for MIMO cases;
(b) an
efficient pole-placement technique is introduced that does not require the characteristic polynomial coefficients, nor the eigenvalues for the original system.
Section 4 covers the critical contributions of this research: the extension of a numerically efficient pole placement technique to the time-varying MZMO dynamics. The Frobenius transformation is again pursued. Pole-placement feedback is then introduced, with a novel generalization of classical formulae from SISO time-invariant to MIMO time-varying systems. We restate that this approach is unique and imposes no restrictions on the dynamics in hand except controllability. Section 5 contains an example to
illustrate the Section 6.
theory.
Conclusions
follow
in
0’
‘i’
-an-2
-a,_,
us first
consider a controllable invariant SISO system given by
I
i=Ax+bu,
(1)
(2)
where 5 (n X 1) is the tranformed state variable vector, a = [a0 al . . . a,_,] are the coefficients of the characteristic polynomial det [zI - A] = Do(z) = z” + u,_~z”-’ + u,_g”-* + * . . +a,z
+ao=o
(3)
of the original system (l), AF (n X n) and BF (n X 1) are the transformed system matrix and the control gain vector, respectively. The passage from (1) to (2) is simply achieved by a similarity transformation using a matrix Q (n X n). That is, 6 = Q-*x,
AF = Q-‘AQ,
BF = Q-‘B.
(4)
For the ultimate objective of pole placement, a state feedback is used over the Frobenius canonical form at hand, as u = K&.
(5)
Following some earlier work of Kalman (1963), Lefschetz (1965) and Tuel (1966) and the final formulation of Luenberger (1967), the transformation matrix Q-’ has the form
(6)
where q1 (1
X
n) is a row vector computed
from the controllability
time-
5
u =A&+++
q, = e:R-’
2. AN OVERVIEW OF POLE PLACEMENT FOR LINEAR SISO SYSTEMS; TIME-VARYING AND TIME-INVARIANT CASES
Let
where x (n X 1) is the state vector, u is the control variable, A (n X n) and B (n X 1) are system and control gain matrices respectively. The intermediate objective to reach is the Frobenius canonical form
R= [B
AB
as (7)
matrix A*B
...
A”-‘B]
(8)
of the system (l), which is taken as full rank. Here e, = [0 0 . . . 0 11’ is a unit vector.
Eigenvalue
The above steps complete the transformation into Frobenius form. It is well known that this form is extremely convenient for executing a feedback stabilization with desired characteristic behavior (i.e. the pole-placement problem). The objective stabilized system is given by either a desired characteristic polynomial or by the eigenvalues (roots) of this polynomial. We treeat both cases below, and present a simplified form of the feedback gain vector KF to be used in the control u = K& of (2). First, we consider the case with a given characteristic equation. A simple substitution of u into (2) yields KF=a-d
(9)
where d = [do d, . . . d,_,] is the vector of the desired characteristic polynomial coefficients parallel to (3). The transformation of (9) into the original state space (i.e. of x), yields the traditional formula of Ackermann (1972, 1977, 1985): K = -q,(A”
+ d,_,A”-’
+ dn_2An-2
+ . . . + d,A + dJ) = -qIDd(A)
(10) where D,(A) denotes the evaluation of the desired characteristic polynomial Dd with the state matrix A. In this formula, if the vector q, is not taken outside the parentheses, a more efficient way of computing the gain vector K is obtained compared with those procedures that are traditionally followed (Franklin and Powell, 1980; Kailath, 1980; Lewis, 1992; MathWorks, 1988). The simple reason for this is that the computation with row vectors is more efficient than that with the full square matrices. The efficient numerical algorithm is straightforward. q ;+I = q;A,
(
n-1
K= - &+, +
c diql+,1.
(11)
i=o
K = -ql ir (A - AjI) j=l
-e;fR-’
,Q (A - A,I).
Similar algorithmic simplifications above discussion of (10) result in , q1=
Ql,
q;+l = &(A
- &I),
i = A(t)x + B(t)u,
(12)
as in the
K = -i+,.
(13)
Equations (12) and (13) constitute the generalized version of Ackermann’s formula, utilizing
(14)
with the state vector x (n X 1) and the scalar control u. The objective is to stabilize the system by means of a linear feedback that enforces a desired characteristic behavior for the state. Note that the eigenvalues of the time-varying dynamic system do not have any meaning regarding its behavior or its stability features. Therefore there is very little on the subject in the literature. One study addressing the issue for time-varying system is that by Nguyen (1987), who uses similar steps as described above for the time-invariant cases. In his treatment the original system is manipulated by a state feedback
(1%
which enforces the plant behavior to be Lyapunov-equivalent to the one with prescribed and fixed eigenvalues. Lyapunov equivalence of two linear systems imply similar stability properties, and also the existence of a Lyapunov transformation from one system to the other that preserves these properties. This manipulation as well utilizes the Frobenius transformation as an intermediate step (Chen, 1984). Essentially, the original system is stabilized by a time-dependent linear state feedback. Let us take a transformation (Chen, 1984) from x to the new state variable 5 (n X 1) via the matrix Q(t): x = Q(t)&
Next, we consider a stabilizing feedback control defined by a set of desired eigenvalues A,; instead of the evaluated coefficients d of the characteristic equation. An alternate form of (10) is
=
the desired poles Aj only, for linear timeinvariant systems. This concludes the poleplacement process for SISO time-invariant systems. The above described treatment can be extended for the general SISO linear timeuarying dynamic system of
u = K(t)x,
= -ezR-‘D,(A),
q;=q,,
1607
assignments for linear MIMO systems
E = Qp’(t)x.
(16)
If Q(t), Q-‘(t) and dQ(t)ldt are continuous and bounded matrices and Q-‘(t) is full rank at the time interval of interest, (to, co), then this transformation is called a Lyapunov transformation. Additionally, the original and transformed systems manifest similar stability properties. The system matrices of the new structure are AF = Q-‘(AQ
- Q),
B, = Q-‘B.
(17)
In the following sections all the time arguments are suppressed for simplicity. The Frobenius canonical form of (2) is again considered, but with time-varying a = a(t) = a,_,(t)]. The time-dependent [a,(t) a,(t) . . .
1608
M. ValiSek and N. Olgac
state transformation matrix Q(r) is determined first, similarly to the work of Silverman (1966) and Nguyen (1987). The matrix Q-’ = rows [sly q2,. . . , qnl is defined by the recursive computation of the rows qi of the matrix Q-‘, qi+ 1 = q;A + Qi,
These formulae state the fact that the poleplacement process can be substantially simplified compared with the technique proposed by Nguyen (1987) owing to the fact that the feedback gain calculations are not done in the intermediate Frobenius domain as in (9). Instead, direct evaluations are performed in the original state space. It also removes the severe restriction in the results of Tsai et al. (1991) that the transformation matrix Q be time-invariant for time-varying systems. This summary concludes the description of the state of the art concerning time-varying SISO cases.
(18)
starting with first row q1 determined as in (7). Note that the controllability matrix R for the time-varying system is different from the time-invariant version (8) (Chen, 1984): R = (ri
r2
...
rn],
rI = B,
q+, = Ari - i;.
(19) Assuming that the above transformation to Frobenius canonical form is of Lyapunov type, we proceed with the desired pole placement. A compact equivalent of Ackermann’s formula in this case is not possible. However, an efficient algorithm like (11) and (13) can be obtained. Leaving the intermediary steps and proofs to ValaSek and Olgac (1995), we give the final form of this algorithm below. If the objective characteristic polynomial is given by its coefficients d;, the feedback gain is selected as
(
K= -
3. POLE PLACEMENT FOR LINEAR TIME-INVARIANT SYSTEMS
Let us now consider the MIMO time-invariant dynamic system of i=Ax+Bu,
starting with q1 = e;fR-‘. If the characteristic polynomial is given by its poles hk (k = 1,. . . , n) then the required feedback gain K is directly computed as
B = [b,
where I q1 = 41,
q;+, = q;(A - AJ) + 4;
1
0 1
X
‘b’ ::: .. . -. .. 0 0 X . .. X .. .. .* X . .. X
X
.. .. .. . ..
0 ..
0
X
A,=
X
X
b2
...
b,]
(24)
are linearly independent. The ultimate objective is to design a linear feedback for this system in accordance with some desired eigenvalues. For the MIMO case at hand, the generalized canonical form, i.e. the corresponding Frobenius form, is defined in Luenberger (1967) as
(21)
K = -de,,
(23)
with state variable x (n X l), control vector u (m x l), m in, and matrices A and B of appropriate dimensions. The first objective is again to reduce the system into a generalized Frobenius canonical form, assuming that the pair [A, B] is controllable, and the m columns bi (n x 1) of the B matrix
(20)
qncl
MIMO
(22)
00 .. 1
. .
Y
X
X
...
x
x
X
X
X
...
x
x
1 0 .. 0
0 1
... 0
0 0
0
..
... ... *. 0
.X ..
.. .. ..
.x
. x. .
X
X
...
x
x
X
1
1
1 (25)
Eigenvalue 0 0
0. 0 ... 0
where e,k (n X 1) are unit vectors with 1 at position rk. From the definition of R-‘,
...
1. 0 0 BF=
0 0
... ...
0
...
0
X
...
X
...... ...... -“I ...... ...... I 0 ... 0 1
.. ..
0 0
L
0 0
0 . 0 ... 0 0
0 0
...
-2
..,
qk@-‘bk
1
... ...
k = 1, . . . , m,
= 1, for k=p for k#p
qkAi-‘b, = 0
0 0
and j
