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Design of Tracking Systems with LQ Optimality and Quadratic Stability
Copyright © IFAC 12th Triennial World Coogress, S ydoey, Australia, 1993
DESIGN OF TRACKING SYSTEMS WITH LQ OPTIMALITY AND QUADRATIC STABILITY T. FujU Department of Control Engineering and Science, Kyushu Institute of Technology, 680-4, Kawazu, Ijzuka, Fukuoka, 82~Japan
Abstract: This paper concerns a tracking problem for linear systems with real uncertain parameters. The objective here is to design a state feedback controller such that 1) the closed-loop system is optimal in the LQ sense for the nominal parameter values, 2) quadratically stable and 3) its output tracks the step reference input asymptotically under the parameter uncertainty. One such controller is obtained here by applying our "Inverse LQ design method" of extended type with quadratic stability requirement incorporated. The attractive feature of the proposed design method lies on one hand in the control structure of the designed tracking system with a low-pass filter placed before the control input, and on another hand in the design parameters with clear physical meaning in relation to this structure. Key Words: Tracking problem, Optimal control, Robust control, Inverse LQ problem, Quadratic stability
of the LQ design, including drastic alleviation of the inherent difficulty with the choice of design parameters in the LQ design (Fujii, 1987). There we have developed a new design method of LQ regulators based on certain results on Inverse LQ problems, called "Inverse LQ design method"
1 INTRODUCTION The linear quadratic (LQ) design method is well known for its attractive feature of robust stability on one hand, and also for its practical difficulty with the choice of design parameters on the other hand. This robustness of LQ regulators is, however, guaranteed only in terms of the stability margin (Anderson and Moore, 1989), and hence with respect to unstructured uncertainty. This does not necessarily mean that the LQ regulator possesses stability robustness to parametric uncertainty as well such as "quadratic stability" (Barmish 1985). In fact, it has been shown via one example (Soroka and Shaked, 1984) that LQ regulators may suffer from poor robustness to parametric uncertainties in spite of well-known large stability margines. From this standpoint, it is desirable to extend this method in such a way that stability robustness to parameter uncertainty can also be taken into consideration in the design process as was treated by Neto et ai, (1992) or Douglas and Athans (1992). Furthermore, it is also desirable to alleviate the practical difficulty with the weight selection. This motivates us to design LQ regulators with quadratic stability, not by the Riccati equation approach as used in the references cited above, but from the viewpoint of Inverse LQ problem. This is because we have succeeded by this approach in considerable simplification
In this paper we deal with such a design problem for the case of optimal servo systems with quadratic stability. More precisely, we treat a tracking problem for a linear system with norm bounded parameter uncertainty such that 1) the closed loop system is LQ optimal in the usual sense for the nominal parameter values, 2) quadratically stable, and 3) its output asymptotically tracks the step reference input. The attractive feature of the design method developed here is three fold; 1) the control structure of the tracking system designed involves a low-pass filter ahead of the control input, being desirable from noise rejection, 2) the design parameters have clear physical meaning: one is the pole parameter specifying the output response approximately, and the other is the gain tuning parameter as a tradeoff parameter between the tracking performance and the magnitude of the control input, 3) all the gains of the feedback controller designed by this method are expressed explicitly in terms of the nominal system parameters and the preceding two design parameters, leading to simplicity of the design algorithm.
435
2 PROBLEM FORMULATION
Then the above design problem of tracking system can be stated essentially as follows:
Consider a linear time-invariant system with structured uncertainty described by x(t)
DESIGN PROBLEM: Design a state feedback control:
= [A + DF Ea] x(t) + [B + DF E b] u(t) yet)
(la) (lb)
= Cx(t)
where x(t) E Rn is the state, u(t) E Rm is the control input, yet) E Rn is the output and F E R!'xq is a matrix of uncertain parameters with its maximum singular value bounded by unity, i.e.,
FE F := {F E Rpxq : IIFII < l}
u
C2)
Robust stability: The closed loop system is quadratically stable under the parameter variations described above.
C3)
=
J =
uT Ru)dt
With F :t 0, the closed loop system (3) and (4) is quadratically stable (Barmish, 1985), i.e., for all allowable F there exists a P > 0 and a (¥ > 0 satisfying
C2)
L(~e) :=
2{{ P[A e + DeFEael~e
-2e; P[Be + De F Eb]/{e~e ::;
-(¥
lI~e 112
C3) The error output z(t) tends to 0 as t for all allowable F.
To define this problem more clearly, we consider a familiar augmented system used in a design problem of tracking systems for a step reference input:
z := y - r = [C 0 1~e
100 ({{ Q~e +
for some Q > 0 and R> O. In this sense, we say that the closed loop system (3) and (4) is nominally LQ optimal.
Robust tracking: Its output yet) approaches r(t) asymptotically as t -+ 00 for all allowable uncertain matrix F.
ee = (Ae + De F Eae)~e + (Be + De F Ebe)U
(4)
With F 0, the state feedback control (4) minimizes a quadratic cost of the form:
Cl)
For this system we consider a state-feedback design problem of tracking systems for a step reference input r(t) E R m with the properties : Nominal LQ optimality: The nominal closed loop system is LQ optimal in the sense described later.
-/{e~e
for the augmented system (3) such that
(2)
Cl)
=
-+ 00
Since C2 implies C3, it suffices to consider those state feedback controls (4) satisfying Cl and C2. With one such state feedback control (4) obtained, the desired tracking system is constructed by using a proportional plus integral state feedback controller of the form:
(3a) (3b)
where
Ae =
[~ ~],
Eae = [Ea
r
0
e
Be
=[ ~ ],
1, De =
[
as shown in Fig. 1, where by
~] U
-1
...c>--.+ (sI-A- M) (B+ L\B)
/{F
and
/{I
are defined
X
Fig. 1 Tracking System (L\A=DFE a, L\B=DFE b) and assume, as usual in this type of tracking problems, that the nominal system (1) with F = o has no zeros at the origin, i.e.,
r
:=
[~ ~] is nonsingular
We should note that the above mentioned nominal LQ optimality and quadratic stability of the closed loop system (3) and (4) are invariant under any state coordinate transformation. Hence for convenience we instead treat the correspond-
(3c) 436
ing design problem of a state feedback control
(7)
B) There exist some V E Rmxm with det V oF 0, E E Rmxm with E = diag{uk} > 0 , and K E Rmxn such that K o = V- 1EV [K
for the following augmented system obtained by the transformation ~e = fX e:
z = [C
0
1fXe
L(P ; Ae, B e, Ke)
(8b)
has a solution P
= [~ ~] ,
Ee = [Ea
Be
=[ ~ ],
0]' De = f-
1
[
~
P
(8c)
= (V K ef
>
>0
(13)
0 of the particular form :
AE-1(V Ko)
+
[~ ~] (14a)
]
for some A E Rmxm and YE R nxn with
(8d)
A> 0, AE = EA, Y > 0
Then the desired K E = [K F K 1] is obtained from Ke by
PRELIMINARIES
To solve the design problem posed above, we derive in this section some preliminary results on conditions for a state feedback control (7) to satisfy the required properties Cl and C2 for the augmented system (8) . 3.1. Conditions for Nominal LQ Optimality The design problem of state feedback controls satisfying the first requirement Cl has been treated in (Fujii, 1987; Fujii et al., 1987) from the viewpoint of the Inverse LQ problem. We first state a basic result obtained there on the Inverse LQ problem, namely an algebraic characterization of LQ optimality of the state feedback control (7) for the nominal (augmented) system:
(10)
(14b)
Furthermore, the R as in (11) is given by:
(9) 3
(12)
and under this condition, the linear matrix inequality
where Ae
1
/
R= VTAV
(15)
Based on this result, we have developed a new design method of LQ regulators, called "Inverse LQ Design Method" , or "ILQ Design Method" for short (see Fujii (1987)). The main idea of this method is to parameterize a control law K e by (12), and then determine the associated parameter matrices V, K, and E such that the corresponding K e is LQ optimal. A theoretical basis for such determination of V, K and E is newly provided here by the following optimality characterization of Ke so parameterized. Proposition 1 The following statements are equivalent.
1)
The state feedback (7) for the system (10) with Ke given by (1 2) minimizes (11) for some Q > 0 and R = V T E- 1 V .
2)
There exists a real matrix Y
>0
satisfying
YAK +AkY [ B~Y + KVAK
Lemma 1 (Fujii et al. , 1987): A control law Ke is LQ optimal in the sens e that th e stat e f eedback (7) for the system (10) minimizes
] <0 (16)
where
J
= 100 (x~ QX e + vT Rv)dt
AK = A - BK, Bv = BV- 1 , K v = VK ,
( 11)
.
D
for some Q > and R > 0 if and only if the following two equivalent conditions hold.
3)
E> KvBv
satisfying
= RK
1 L(P; Ae , B e, K o) := P(2BoKo - Ae) 1 T +(2BoK e - A e) P
K v Bv - (K v Bv)
where
=Q 437
T
The following conditions hold.
A) There exists some P > 0, Q > 0 and R> 0 B~P
=E -
+ (/{vBv)T
(17a)
ReA(A) < 0
(17b)
116(s/ - A)-1 Blloo < 1
(17c)
We call such a tracking system ILQ Servo System for convenience, and present in the sequel a practical design algorithm of V, K and E so as to satisfy the nominal LQ optimality condition (17) as well as the small gain condition (19).
(see Appendix for the proof).
Remark 1 Let (16) hold for some V, E, K, and Y. Then it remains to hold even if we replace E alone by any El with El ~ E. This together with Lemma 1 implies that we can increase E alone as much as we like without violating the LQ optimality of Ke given by (l2), once we can find some V, E and K satisfying the condition :1). In addition, considering the extreme case of E ..... 00 in (l7b) yields the stability of AK as a necessary condition for LQ optimality of Ke . Conversely, the stability of AK ensures (17) for sufficiently large values ofE, as can be seen from the form of (17).
4 .1. Control Structure of ILQ Servo System
First of all we investigate the control structure of ILQ servo system and clarify its special control structure together with the associated roles played by the parameter matrices E and K. For this purpose it is convenient to define normalized gain matrices for K F and K /, denoted by K~ and KJ as
3.2. Conditions for Quadratic Stability
[K~
The condition C2 for quadratic stability of the uncertain system (8) with the state feedback (7) follows by direct application of a well-known result on quadratic stability , known as a small gain condition (Khargonekar et al., 1990).
KJ 1= [K I 1r- l
(21)
so that by definition of r we have K~A
+ KJC = K ,
K~B
=I
(22)
The terminology normalized gain comes from the obvious relation following from (20) and (21):
Proposition 2 The uncertain system (8) with the state feedback (7) is quadratically stable if and only if the following two conditions hold.
is
stable .
IIGe(s)lIoo < 1,
meaning that the normalized gains correspond to those gains KF and K/ for E = I. In addition, by solving (22) for K~ and KJ we can obtain their analytical expressions in terms of K as follows:
(18) (19)
Ge(s) := (Ee - EbKe)(sI -
MAIN RESULTS (24a)
In the previous section we derived some basic results on the design of a state feedback gain Ke for the augmented system (8) such that the resulting closed loop system is both Cl) nominally LQ optimal and C2) quadratically stable. Namely, 1)
2)
(24b) In accordance with this naming we call E a gain tuning parameter in view of (23). The next important step is to rewrite the uncertain system (1) as a system with an uncertain feedback from the controlled variable z to the disturbance w :
The requirement Cl led us to a simple parameterization of Ke in the form of (12) with the constraint (17) among its parameter matrices V, K and E (Proposition 1).
z
The requirement C2 imposed additionally the small gain condition (19) on Ke (Proposition 2); note that the nominal closed loop stability condition (18) is implied obviously by the nominal LQ optimality condition (17).
w
K/
1=
V-1EV [K
I
1r- l
(2Sa)
+ Eb U
(2Sb) (2Sc)
Eax Fz
The control structure of ILQ servo system is then clarified as follows . Proposition 3 The ILQ servo system has a control structure as shown in Fig . 2 with a first-
Our aim in this section is to integrate these basic results into a design method of the desired tracking system satisfying the requirements Cl and C2. In view of 1) stated above, we shall hereafter pay attention to a particular tracking system with the control input (S), in which KF and K/ are determined from (9) based on the LQ control law Ke of the form (12), i.e., [KF
Ax+Bu+Dw
order filter of the form:
ahead of the plant, whose input consists of a state feedback (-Kx) , a reference feedforward (KJr) and a disturbance feedforward (-K~Dw) with gains KJ and K~ given by (24).
(20) 438
r------------IFI4---------.. . . w
z
hr---------------. r------~ Ebt----<>
r
'----------fKI4---------' Fig. 2
The Control Structure of ILQ Servo System
Proof Differentiating (5) and substituting the equivalent expression of the system (1) as given by (25) together with (23), and then using (22) yields
u=
V-1EV(-u - Kz
+ KJr -
2)
Wd(S)
KJ..Dw)
r I·················································································; ·1
for the ILQ servo system. Noting the diagonal structure of E = diag{O"k}, this relation obviously yields the control structure stated above and completes the proof.
,[ ,
An interesting characteristic of this control structure lies of course in the low pass filter H(s) as a series compensator. The significance of existence of such a filter is clear from the viewpoint of noise or control input, that is, the role of this filter is to reject at the control input high frequency noise involved in the feedback signal or regulate the magnitude of the control input by changing its bandwidth, a very reasonable and desirable control structure. In addition, the bandwidth of this filter and maybe the magnitude of the control input as well can be regulated by the gain tuning parameter E, a key role of E, which is one of the most attractive features of this control system.
!. . •••••••••• .• •• ••• ••••• •••• •••••• _ ...... .... .. .. . ... ...... ........ .................. :
Fig. 3 Asymptotic ILQ Servo System
Proof
Property 1) is obvious from the structure of ILQ servo system shown in Fig. 2. Property 2) follows easily from the associated characteristic equation:
11 + K(sl = IV-1diag
A)-1 BV- 1H(s)VI
{:k + 1}
V +K(sl - A)-l
BI
xlfI ~I=o + k=l u.
4.2. Asymptotic Properties of the nominal ILQ Servo System
1
Remark 2 The convergence of W~r(s) into Wis) as shown above can be verified also in a stronger sense of Loo under stability assumption
In order to find the role of another design parameter K, we need to state an asymptotic property of the nominal ILQ servo system that stems from the special control structure of ILQ servo system stated above.
of AK, namely, the following asymptotic properly holds.
Proposition 4 As all diagonal elements of E tend to infinity, denoted by E - 00, the nominal ILQ servo system, or the tracking system of Fig. 2 with F = 0, has the following asymptotic properties regarding its closed loop transfer function and its poles.
1)
The n closed loop poles tend to the eigenvalues of AK, and the others tend to those of -E.
(We omit the proof for short of space.)
We see from this result that the role of the design parameter K is to determine the basic closed loop characteristic of the nominal ILQ servo system, in the sense that it determines the nominal closed loop transfer function W~r(s) asymptotically as E increases (hence we call it a basic feedback gain parameter). In fact, this role of K provides an important basis for the determination of K in ILQ design so as to yield a desired
The nominal ILQ closed loop transfer matm: from r to y, W~r(s), tends to that of a system shown in Fig. 3, Wd(s), or equivalently,
W~r(s) - Wd(s) := C(sl - AK )-1 BKJ (26) 439
output response.
where
4.3. Determination of K
N..,.-'I'
In view of the asymptotic property (26), we shall determine K so as to yield a desired nominal output response of ILQ servo system, or equivalently, a desired Wd(S), As a desired Wd(S) we consider here, from the practical viewpoint, a diagonal rational matrix of the form:
Mo
= diag{
(see Appendix for the proof) Combining Propositions 4 and 5 yields the following result as a basis for achieving the output response specification of ILQ servo system, which is one of the most attractive features of ILQ servo design method. Theorem 1 Under the assumption above, let us determine K by (32) and K~, KJ by (33) for the ILQ servo system as shown in Fig. 2. Then the nominal step response of the ILQ servo system approaches, as E --+ 00, that of a system with the transfer matrix:
leading to a decoupled output response of practical importance. Our aim is, therefore, to obtain K satisfying this equation for some polynomial functions {
Wd () S
Let us denote the i-th row of C by ci(l :5 i :5 m) and define the following indices di and matrix M:
::I O}
],
cm ?/lm (A)
(27)
~ := min{k I ciA k- 1 B
[Cl?/l~:(A)
:=
. {
(34)
This result suggests us to use Si (1 :5 i :5 m) in (30) as a design parameter for specifying the i-th nominal output response of ILQ servo system, and {Ui} as tradeoff parameters between the output tracking property stated above and the bandwidth of the internal filter H(s), or the magnitude of the control input. Since the design parameter {ud are the poles of Wd(s), we hereafter denote them by
(1:5 i :5 m)
(28) (29)
S= {sd
and make the following assumption. Assumption The nominal system (1) is minimum phase and decouplable by state feedback i.e., det M ::I 0 (Gilbert, 1969).
and call them pole parameters for convenience.
We then define two polynomials:
Since all the design parameter matrices V, K, and E must be chosen so as to meet the design requirements Cl and C2, or equivalently the nominal LQ optimality condition (17) and the small gain condition (19), so must be chosen Sand E. In regard to the former condition, we can propose a simple method as follows:
4.4. Determination of Sand E
d;
= (S-Si)d; = Eausk
1:5
i:5 m (30)
k=O
,/Ji(S)
= {
I:l
Pik Sk
1:5 i :5 m
Method I Choose any real Si < 0 (1 :5 i :5 m). Restrict E in the form E = u I and find a minimum possible value of u satisfying (17), denoted by!!., by using the so-called bisection method. Then any Ui ~ !!. (1 :5 i :5 m) can be chosen without violating the nominal LQ optimality as stated in Remark 1.
k=O
(31) where Si, 1 :5 i :5 m are all real negative values. Proposition 5 With the assumption and the definition of
In case of the latter condition, however, we cannot choose Sand E independently in general, since K and E are related in a complex way to this condition. To make this condition look more transparent, let us convert it by the state coordinate transformation:
(32)
Furthermore, the corresponding normalized gain matrices K~ and KJ are given by
Xe
(33) 440
=
[-E~K -~v]
[: ],
Ev := V-1EV
By simple calculation we can see that Ge(s) as defined in (19) is the transfer matrix from w to z of the nominal ILQ servo system of Fig. 2, the state equation of which is given by
x
Ax+Bu+Dw
u
Ev(-u - I{x
the following design method of ILQ servo system with both quadratic stability and the nominal LQ optimality, which we call ILQ robust servo design method. Theorem 2 Assume the nominal system (J) is minimum phase and is decouplable by state feedback, or det M '# O. Then the ILQ servo system meeting the requirements Cl and C2 can be designed by the following procedure:
+ K~Dw)
Therefore, the condition (19) can be interpreted as the small gain condition for the nominal ILQ servo system of Fig. 2 with the input wand the output z . Based on this observation, we can propose the following method to determine Sand E independently.
Step 1. Determine the pole parameters S = {sd by Method Il. Step 2. Determine by Method I the lower bound Q. of the tuning parameter that satisfies (17) for the S = {Si} obtained above.
Method 11 Step1) Choose any real Si < 0 (1 ~ i ~ m) so as to satisfy the small gain condition on the transfer matrix Wz~ (s):
Step 3. Determine the tuning parameters E diag{ O';} with O'i ~ Q. (1 ~ i ~ m) by Step 2 of Method Il.
where Wz~(s) is the transfer matrix from w to z of the nominal ILQ servo system of Fig. 2 as obtained in the limit of E --> 00, and is written easily by
Step
K~ and KJ by (33) in terms of the system matrices as well as the pole parameters S obtained above .
4. Determine the normalized gains
Step 5. Give the configuration of the ILQ servo system as in Fig. 4.
where Furthermore, the nominal step response of this tracking system has the asymptotic property as stated in Theorem 1.
Ga(s) = Ea(sI - AK )-1(1 - BK~)D Gb(s) = Eb [K(sI - A K )-l(1 - BK~)
r
+ K~l
D
e
Fig. 4 ILQ Servo System Remark 3 A fundamental idea behind Step 1 is to determine the design parameter S such that the resulting ILQ servo system is quadratically stable in the limit of E --> 00. Therefore, if this step succeeds, we can always obtain the desired ILQ servo system by choosing sufficiently large values of E (see also Remark 1). The specific values of such E are then determined at Step 3 without violating the nominal LQ optimality requirement. However, if this step fails, we must determine Sand E dependently so as to satisfy the small gain condition (19). If, in addition, such Sand E cannot be found, we have to utilize the freedom available in the choice of V before giving up the requirement C2.
Step2) Choose O'i > 0 (1 ~ i ~ m) so as to satisfy the small gain condition (19) on the transfer matrix Ge(s). If no such O'i can be chosen, then go back to Step 1 and rep eat the process until the desired Sand E are obtained. Up to this point , no mention has been made about the choice of the remaining parameter V . As is clear from the foregoing development, this matrix seems to have no essential effect on the two requirements Cl and C2. Therefore we set V = I for simplicity of the design procedure. 4.5. ILQ Robust Servo Design Method Summarizing the foregoing discussion , we obtain 441
5
statement 1) holds iff (13) has a solution P > 0 of the form (14) with A = E- 1 , or equivalently, for some P > 0 of this form, the positive definite matrix
CONCLUSION
By taking the Inverse LQ approach, we have develpoted an optimal servo design method in which quadratic stability can also be taken into consideration in the design process of desired tracking systems. In the course of this development, we clarified the control structure of the tracking system of our interest, i.e., ILQ servo system, as well as the physical meaning of its design parameters in relation to this structure. By making full use of this structure, we were able to obtain not only the simple design algorithm but also the simplified derivation of the existing ILQ design method as a side effect. One important subject for future study along this line is to clarify the role of the parameter V, especially in relation to robust stabilization, together with its effective way of determination. 6
P := TT PT,
V~l]
T := [_IK
of a block diagonal form P = diag{Y, EA} diag{Y, I} by the definition, satisfies
L(P;T- 1 A eT,T- 1 B e, Ke T ) > 0 Substituting this P together with (8c) and (12) into this inequality and rewriting yields (16), or equivalently
D>O and yA+J.\Ty+yB1Fy+cTc<0 The latter inequality is equivalent to the conditions (17b) and (17 c) as was shown in Khargonekar et al.(1990). This completes the proof.
Proof of Proposition 5
REFERENCE
With K given by (32), we first derive (33) by showing that these normalized gains satisfy their definitive equation (22). Substituting (32) and (33) into (22) we see that this is true if
Anderson. B. D. O. and J. B. Moore (1990). Optimal Control: Linear Quadratic Method, Prentice-Hall, New Jersey. Barmish, B. R. (1985). Necessary and Sufficient Conditions for Quadratic Stabilizability of an Uncertain System. J. Optimiz. Theory Appl. 46, 399-408. Douglas, J., and M. Athans (1992) . R0bust Linear Quadratic Designs with Respect to Parameter Uncertainty. Proc. 1992 American Control Conference, 2905-2910. Fujii, T. (1987) . A new approach to LQ design from the viewpoint of the inverse regulator problem. IEEE Trans. Aut. Control AC-32,
N",A + MoC
= N.p,
N",B
or equivalently for each i, 1 ~ i CitPi(A)A+~i(O)Ci
= ci~i(A),
~
=M
m
citPi(A)B
= Mi
holds. The left relation follows immediately by multiplying the left relation of (31) by sand substituting s = A, and finally left-multiplying Ci. The right relation follows by substituting the right relation of (31) and noting the definitions of di and M i . We now remains to show that (32) and the latter part of (33) satisfy (27). Differentiating (lb) and substituting the nominal system equation (la) with F = 0 yields by definition of di :
99~1004 .
Fujii, T., Y. Nishimura, T. Shimomura and S. Kawarabayashi (1987). A Practical Approach to LQ design and its application to engine control. Proc. 10th IFAC World Congress, pp. 253-258, Munich. Gilbert, E. G. (1969). The decoupling of multivariable systems by state feedback. SIAM J. Ctrl. and Opt., 7, 50-63. Khargonekar, P. P., I. R. Petersen and K. Zhou (1990). Robust Stabilization of Uncertain Linear Systems: Quadratic Stabilizability and Hoo Control Theory. IEEE funs. Aut. Control AC-35, 356-361. Neto, A. T., J. M. Dion and L. Dugard (1992) . Robustness Bounds for LQ Regulators. IEEE Trans. Aut. Control AC-37, 1373-1377. Soroka, E., and U. Shaked (1984). On The Robustness of LQ Regulators. IEEE Trans. Aut. Control AC-29, 664-665.
dkYi(t)
dr
= ciAkx(t)
crI'yi(t)
(35)
dtd.
where Yi(t) denotes the i-th component of y(t). Taking the Laplace transform of these equations, multiplying each of them by O'ik (0 ~ k ~ di 1) and finally summing up these equations from k = 0 to k = di yields by definition of ~i(S): ~i(S)Yi(S)
=
Ci~i(A)X(s)
+ MiU(s)
Substituting the control law U(s) = -KX(s) + KJR(s) with K = M- 1 N.p and KJ = M- 1 Mo yields i = 1,"',m
APPENDIX: PROOFS
By definition of Wd(S) as in (26) , this is equivalent to saying that (27) holds as required.
Proof of Proposition 1 As is clear from Lemma 1 and Remark 1, the 442
DESIGN OF TRACKING SYSTEMS WITH LQ OPTIMALITY AND QUADRATIC STABILITY T. FujU Department of Control Engineering and Science, Kyushu Institute of Technology, 680-4, Kawazu, Ijzuka, Fukuoka, 82~Japan
Abstract: This paper concerns a tracking problem for linear systems with real uncertain parameters. The objective here is to design a state feedback controller such that 1) the closed-loop system is optimal in the LQ sense for the nominal parameter values, 2) quadratically stable and 3) its output tracks the step reference input asymptotically under the parameter uncertainty. One such controller is obtained here by applying our "Inverse LQ design method" of extended type with quadratic stability requirement incorporated. The attractive feature of the proposed design method lies on one hand in the control structure of the designed tracking system with a low-pass filter placed before the control input, and on another hand in the design parameters with clear physical meaning in relation to this structure. Key Words: Tracking problem, Optimal control, Robust control, Inverse LQ problem, Quadratic stability
of the LQ design, including drastic alleviation of the inherent difficulty with the choice of design parameters in the LQ design (Fujii, 1987). There we have developed a new design method of LQ regulators based on certain results on Inverse LQ problems, called "Inverse LQ design method"
1 INTRODUCTION The linear quadratic (LQ) design method is well known for its attractive feature of robust stability on one hand, and also for its practical difficulty with the choice of design parameters on the other hand. This robustness of LQ regulators is, however, guaranteed only in terms of the stability margin (Anderson and Moore, 1989), and hence with respect to unstructured uncertainty. This does not necessarily mean that the LQ regulator possesses stability robustness to parametric uncertainty as well such as "quadratic stability" (Barmish 1985). In fact, it has been shown via one example (Soroka and Shaked, 1984) that LQ regulators may suffer from poor robustness to parametric uncertainties in spite of well-known large stability margines. From this standpoint, it is desirable to extend this method in such a way that stability robustness to parameter uncertainty can also be taken into consideration in the design process as was treated by Neto et ai, (1992) or Douglas and Athans (1992). Furthermore, it is also desirable to alleviate the practical difficulty with the weight selection. This motivates us to design LQ regulators with quadratic stability, not by the Riccati equation approach as used in the references cited above, but from the viewpoint of Inverse LQ problem. This is because we have succeeded by this approach in considerable simplification
In this paper we deal with such a design problem for the case of optimal servo systems with quadratic stability. More precisely, we treat a tracking problem for a linear system with norm bounded parameter uncertainty such that 1) the closed loop system is LQ optimal in the usual sense for the nominal parameter values, 2) quadratically stable, and 3) its output asymptotically tracks the step reference input. The attractive feature of the design method developed here is three fold; 1) the control structure of the tracking system designed involves a low-pass filter ahead of the control input, being desirable from noise rejection, 2) the design parameters have clear physical meaning: one is the pole parameter specifying the output response approximately, and the other is the gain tuning parameter as a tradeoff parameter between the tracking performance and the magnitude of the control input, 3) all the gains of the feedback controller designed by this method are expressed explicitly in terms of the nominal system parameters and the preceding two design parameters, leading to simplicity of the design algorithm.
435
2 PROBLEM FORMULATION
Then the above design problem of tracking system can be stated essentially as follows:
Consider a linear time-invariant system with structured uncertainty described by x(t)
DESIGN PROBLEM: Design a state feedback control:
= [A + DF Ea] x(t) + [B + DF E b] u(t) yet)
(la) (lb)
= Cx(t)
where x(t) E Rn is the state, u(t) E Rm is the control input, yet) E Rn is the output and F E R!'xq is a matrix of uncertain parameters with its maximum singular value bounded by unity, i.e.,
FE F := {F E Rpxq : IIFII < l}
u
C2)
Robust stability: The closed loop system is quadratically stable under the parameter variations described above.
C3)
=
J =
uT Ru)dt
With F :t 0, the closed loop system (3) and (4) is quadratically stable (Barmish, 1985), i.e., for all allowable F there exists a P > 0 and a (¥ > 0 satisfying
C2)
L(~e) :=
2{{ P[A e + DeFEael~e
-2e; P[Be + De F Eb]/{e~e ::;
-(¥
lI~e 112
C3) The error output z(t) tends to 0 as t for all allowable F.
To define this problem more clearly, we consider a familiar augmented system used in a design problem of tracking systems for a step reference input:
z := y - r = [C 0 1~e
100 ({{ Q~e +
for some Q > 0 and R> O. In this sense, we say that the closed loop system (3) and (4) is nominally LQ optimal.
Robust tracking: Its output yet) approaches r(t) asymptotically as t -+ 00 for all allowable uncertain matrix F.
ee = (Ae + De F Eae)~e + (Be + De F Ebe)U
(4)
With F 0, the state feedback control (4) minimizes a quadratic cost of the form:
Cl)
For this system we consider a state-feedback design problem of tracking systems for a step reference input r(t) E R m with the properties : Nominal LQ optimality: The nominal closed loop system is LQ optimal in the sense described later.
-/{e~e
for the augmented system (3) such that
(2)
Cl)
=
-+ 00
Since C2 implies C3, it suffices to consider those state feedback controls (4) satisfying Cl and C2. With one such state feedback control (4) obtained, the desired tracking system is constructed by using a proportional plus integral state feedback controller of the form:
(3a) (3b)
where
Ae =
[~ ~],
Eae = [Ea
r
0
e
Be
=[ ~ ],
1, De =
[
as shown in Fig. 1, where by
~] U
-1
...c>--.+ (sI-A- M) (B+ L\B)
/{F
and
/{I
are defined
X
Fig. 1 Tracking System (L\A=DFE a, L\B=DFE b) and assume, as usual in this type of tracking problems, that the nominal system (1) with F = o has no zeros at the origin, i.e.,
r
:=
[~ ~] is nonsingular
We should note that the above mentioned nominal LQ optimality and quadratic stability of the closed loop system (3) and (4) are invariant under any state coordinate transformation. Hence for convenience we instead treat the correspond-
(3c) 436
ing design problem of a state feedback control
(7)
B) There exist some V E Rmxm with det V oF 0, E E Rmxm with E = diag{uk} > 0 , and K E Rmxn such that K o = V- 1EV [K
for the following augmented system obtained by the transformation ~e = fX e:
z = [C
0
1fXe
L(P ; Ae, B e, Ke)
(8b)
has a solution P
= [~ ~] ,
Ee = [Ea
Be
=[ ~ ],
0]' De = f-
1
[
~
P
(8c)
= (V K ef
>
>0
(13)
0 of the particular form :
AE-1(V Ko)
+
[~ ~] (14a)
]
for some A E Rmxm and YE R nxn with
(8d)
A> 0, AE = EA, Y > 0
Then the desired K E = [K F K 1] is obtained from Ke by
PRELIMINARIES
To solve the design problem posed above, we derive in this section some preliminary results on conditions for a state feedback control (7) to satisfy the required properties Cl and C2 for the augmented system (8) . 3.1. Conditions for Nominal LQ Optimality The design problem of state feedback controls satisfying the first requirement Cl has been treated in (Fujii, 1987; Fujii et al., 1987) from the viewpoint of the Inverse LQ problem. We first state a basic result obtained there on the Inverse LQ problem, namely an algebraic characterization of LQ optimality of the state feedback control (7) for the nominal (augmented) system:
(10)
(14b)
Furthermore, the R as in (11) is given by:
(9) 3
(12)
and under this condition, the linear matrix inequality
where Ae
1
/
R= VTAV
(15)
Based on this result, we have developed a new design method of LQ regulators, called "Inverse LQ Design Method" , or "ILQ Design Method" for short (see Fujii (1987)). The main idea of this method is to parameterize a control law K e by (12), and then determine the associated parameter matrices V, K, and E such that the corresponding K e is LQ optimal. A theoretical basis for such determination of V, K and E is newly provided here by the following optimality characterization of Ke so parameterized. Proposition 1 The following statements are equivalent.
1)
The state feedback (7) for the system (10) with Ke given by (1 2) minimizes (11) for some Q > 0 and R = V T E- 1 V .
2)
There exists a real matrix Y
>0
satisfying
YAK +AkY [ B~Y + KVAK
Lemma 1 (Fujii et al. , 1987): A control law Ke is LQ optimal in the sens e that th e stat e f eedback (7) for the system (10) minimizes
] <0 (16)
where
J
= 100 (x~ QX e + vT Rv)dt
AK = A - BK, Bv = BV- 1 , K v = VK ,
( 11)
.
D
for some Q > and R > 0 if and only if the following two equivalent conditions hold.
3)
E> KvBv
satisfying
= RK
1 L(P; Ae , B e, K o) := P(2BoKo - Ae) 1 T +(2BoK e - A e) P
K v Bv - (K v Bv)
where
=Q 437
T
The following conditions hold.
A) There exists some P > 0, Q > 0 and R> 0 B~P
=E -
+ (/{vBv)T
(17a)
ReA(A) < 0
(17b)
116(s/ - A)-1 Blloo < 1
(17c)
We call such a tracking system ILQ Servo System for convenience, and present in the sequel a practical design algorithm of V, K and E so as to satisfy the nominal LQ optimality condition (17) as well as the small gain condition (19).
(see Appendix for the proof).
Remark 1 Let (16) hold for some V, E, K, and Y. Then it remains to hold even if we replace E alone by any El with El ~ E. This together with Lemma 1 implies that we can increase E alone as much as we like without violating the LQ optimality of Ke given by (l2), once we can find some V, E and K satisfying the condition :1). In addition, considering the extreme case of E ..... 00 in (l7b) yields the stability of AK as a necessary condition for LQ optimality of Ke . Conversely, the stability of AK ensures (17) for sufficiently large values ofE, as can be seen from the form of (17).
4 .1. Control Structure of ILQ Servo System
First of all we investigate the control structure of ILQ servo system and clarify its special control structure together with the associated roles played by the parameter matrices E and K. For this purpose it is convenient to define normalized gain matrices for K F and K /, denoted by K~ and KJ as
3.2. Conditions for Quadratic Stability
[K~
The condition C2 for quadratic stability of the uncertain system (8) with the state feedback (7) follows by direct application of a well-known result on quadratic stability , known as a small gain condition (Khargonekar et al., 1990).
KJ 1= [K I 1r- l
(21)
so that by definition of r we have K~A
+ KJC = K ,
K~B
=I
(22)
The terminology normalized gain comes from the obvious relation following from (20) and (21):
Proposition 2 The uncertain system (8) with the state feedback (7) is quadratically stable if and only if the following two conditions hold.
is
stable .
IIGe(s)lIoo < 1,
meaning that the normalized gains correspond to those gains KF and K/ for E = I. In addition, by solving (22) for K~ and KJ we can obtain their analytical expressions in terms of K as follows:
(18) (19)
Ge(s) := (Ee - EbKe)(sI -
MAIN RESULTS (24a)
In the previous section we derived some basic results on the design of a state feedback gain Ke for the augmented system (8) such that the resulting closed loop system is both Cl) nominally LQ optimal and C2) quadratically stable. Namely, 1)
2)
(24b) In accordance with this naming we call E a gain tuning parameter in view of (23). The next important step is to rewrite the uncertain system (1) as a system with an uncertain feedback from the controlled variable z to the disturbance w :
The requirement Cl led us to a simple parameterization of Ke in the form of (12) with the constraint (17) among its parameter matrices V, K and E (Proposition 1).
z
The requirement C2 imposed additionally the small gain condition (19) on Ke (Proposition 2); note that the nominal closed loop stability condition (18) is implied obviously by the nominal LQ optimality condition (17).
w
K/
1=
V-1EV [K
I
1r- l
(2Sa)
+ Eb U
(2Sb) (2Sc)
Eax Fz
The control structure of ILQ servo system is then clarified as follows . Proposition 3 The ILQ servo system has a control structure as shown in Fig . 2 with a first-
Our aim in this section is to integrate these basic results into a design method of the desired tracking system satisfying the requirements Cl and C2. In view of 1) stated above, we shall hereafter pay attention to a particular tracking system with the control input (S), in which KF and K/ are determined from (9) based on the LQ control law Ke of the form (12), i.e., [KF
Ax+Bu+Dw
order filter of the form:
ahead of the plant, whose input consists of a state feedback (-Kx) , a reference feedforward (KJr) and a disturbance feedforward (-K~Dw) with gains KJ and K~ given by (24).
(20) 438
r------------IFI4---------.. . . w
z
hr---------------. r------~ Ebt----<>
r
'----------fKI4---------' Fig. 2
The Control Structure of ILQ Servo System
Proof Differentiating (5) and substituting the equivalent expression of the system (1) as given by (25) together with (23), and then using (22) yields
u=
V-1EV(-u - Kz
+ KJr -
2)
Wd(S)
KJ..Dw)
r I·················································································; ·1
for the ILQ servo system. Noting the diagonal structure of E = diag{O"k}, this relation obviously yields the control structure stated above and completes the proof.
,[ ,
An interesting characteristic of this control structure lies of course in the low pass filter H(s) as a series compensator. The significance of existence of such a filter is clear from the viewpoint of noise or control input, that is, the role of this filter is to reject at the control input high frequency noise involved in the feedback signal or regulate the magnitude of the control input by changing its bandwidth, a very reasonable and desirable control structure. In addition, the bandwidth of this filter and maybe the magnitude of the control input as well can be regulated by the gain tuning parameter E, a key role of E, which is one of the most attractive features of this control system.
!. . •••••••••• .• •• ••• ••••• •••• •••••• _ ...... .... .. .. . ... ...... ........ .................. :
Fig. 3 Asymptotic ILQ Servo System
Proof
Property 1) is obvious from the structure of ILQ servo system shown in Fig. 2. Property 2) follows easily from the associated characteristic equation:
11 + K(sl = IV-1diag
A)-1 BV- 1H(s)VI
{:k + 1}
V +K(sl - A)-l
BI
xlfI ~I=o + k=l u.
4.2. Asymptotic Properties of the nominal ILQ Servo System
1
Remark 2 The convergence of W~r(s) into Wis) as shown above can be verified also in a stronger sense of Loo under stability assumption
In order to find the role of another design parameter K, we need to state an asymptotic property of the nominal ILQ servo system that stems from the special control structure of ILQ servo system stated above.
of AK, namely, the following asymptotic properly holds.
Proposition 4 As all diagonal elements of E tend to infinity, denoted by E - 00, the nominal ILQ servo system, or the tracking system of Fig. 2 with F = 0, has the following asymptotic properties regarding its closed loop transfer function and its poles.
1)
The n closed loop poles tend to the eigenvalues of AK, and the others tend to those of -E.
(We omit the proof for short of space.)
We see from this result that the role of the design parameter K is to determine the basic closed loop characteristic of the nominal ILQ servo system, in the sense that it determines the nominal closed loop transfer function W~r(s) asymptotically as E increases (hence we call it a basic feedback gain parameter). In fact, this role of K provides an important basis for the determination of K in ILQ design so as to yield a desired
The nominal ILQ closed loop transfer matm: from r to y, W~r(s), tends to that of a system shown in Fig. 3, Wd(s), or equivalently,
W~r(s) - Wd(s) := C(sl - AK )-1 BKJ (26) 439
output response.
where
4.3. Determination of K
N..,.-'I'
In view of the asymptotic property (26), we shall determine K so as to yield a desired nominal output response of ILQ servo system, or equivalently, a desired Wd(S), As a desired Wd(S) we consider here, from the practical viewpoint, a diagonal rational matrix of the form:
Mo
= diag{
(see Appendix for the proof) Combining Propositions 4 and 5 yields the following result as a basis for achieving the output response specification of ILQ servo system, which is one of the most attractive features of ILQ servo design method. Theorem 1 Under the assumption above, let us determine K by (32) and K~, KJ by (33) for the ILQ servo system as shown in Fig. 2. Then the nominal step response of the ILQ servo system approaches, as E --+ 00, that of a system with the transfer matrix:
leading to a decoupled output response of practical importance. Our aim is, therefore, to obtain K satisfying this equation for some polynomial functions {
Wd () S
Let us denote the i-th row of C by ci(l :5 i :5 m) and define the following indices di and matrix M:
::I O}
],
cm ?/lm (A)
(27)
~ := min{k I ciA k- 1 B
[Cl?/l~:(A)
:=
. {
(34)
This result suggests us to use Si (1 :5 i :5 m) in (30) as a design parameter for specifying the i-th nominal output response of ILQ servo system, and {Ui} as tradeoff parameters between the output tracking property stated above and the bandwidth of the internal filter H(s), or the magnitude of the control input. Since the design parameter {ud are the poles of Wd(s), we hereafter denote them by
(1:5 i :5 m)
(28) (29)
S= {sd
and make the following assumption. Assumption The nominal system (1) is minimum phase and decouplable by state feedback i.e., det M ::I 0 (Gilbert, 1969).
and call them pole parameters for convenience.
We then define two polynomials:
Since all the design parameter matrices V, K, and E must be chosen so as to meet the design requirements Cl and C2, or equivalently the nominal LQ optimality condition (17) and the small gain condition (19), so must be chosen Sand E. In regard to the former condition, we can propose a simple method as follows:
4.4. Determination of Sand E
d;
= (S-Si)d; = Eausk
1:5
i:5 m (30)
k=O
,/Ji(S)
= {
I:l
Pik Sk
1:5 i :5 m
Method I Choose any real Si < 0 (1 :5 i :5 m). Restrict E in the form E = u I and find a minimum possible value of u satisfying (17), denoted by!!., by using the so-called bisection method. Then any Ui ~ !!. (1 :5 i :5 m) can be chosen without violating the nominal LQ optimality as stated in Remark 1.
k=O
(31) where Si, 1 :5 i :5 m are all real negative values. Proposition 5 With the assumption and the definition of
In case of the latter condition, however, we cannot choose Sand E independently in general, since K and E are related in a complex way to this condition. To make this condition look more transparent, let us convert it by the state coordinate transformation:
(32)
Furthermore, the corresponding normalized gain matrices K~ and KJ are given by
Xe
(33) 440
=
[-E~K -~v]
[: ],
Ev := V-1EV
By simple calculation we can see that Ge(s) as defined in (19) is the transfer matrix from w to z of the nominal ILQ servo system of Fig. 2, the state equation of which is given by
x
Ax+Bu+Dw
u
Ev(-u - I{x
the following design method of ILQ servo system with both quadratic stability and the nominal LQ optimality, which we call ILQ robust servo design method. Theorem 2 Assume the nominal system (J) is minimum phase and is decouplable by state feedback, or det M '# O. Then the ILQ servo system meeting the requirements Cl and C2 can be designed by the following procedure:
+ K~Dw)
Therefore, the condition (19) can be interpreted as the small gain condition for the nominal ILQ servo system of Fig. 2 with the input wand the output z . Based on this observation, we can propose the following method to determine Sand E independently.
Step 1. Determine the pole parameters S = {sd by Method Il. Step 2. Determine by Method I the lower bound Q. of the tuning parameter that satisfies (17) for the S = {Si} obtained above.
Method 11 Step1) Choose any real Si < 0 (1 ~ i ~ m) so as to satisfy the small gain condition on the transfer matrix Wz~ (s):
Step 3. Determine the tuning parameters E diag{ O';} with O'i ~ Q. (1 ~ i ~ m) by Step 2 of Method Il.
where Wz~(s) is the transfer matrix from w to z of the nominal ILQ servo system of Fig. 2 as obtained in the limit of E --> 00, and is written easily by
Step
K~ and KJ by (33) in terms of the system matrices as well as the pole parameters S obtained above .
4. Determine the normalized gains
Step 5. Give the configuration of the ILQ servo system as in Fig. 4.
where Furthermore, the nominal step response of this tracking system has the asymptotic property as stated in Theorem 1.
Ga(s) = Ea(sI - AK )-1(1 - BK~)D Gb(s) = Eb [K(sI - A K )-l(1 - BK~)
r
+ K~l
D
e
Fig. 4 ILQ Servo System Remark 3 A fundamental idea behind Step 1 is to determine the design parameter S such that the resulting ILQ servo system is quadratically stable in the limit of E --> 00. Therefore, if this step succeeds, we can always obtain the desired ILQ servo system by choosing sufficiently large values of E (see also Remark 1). The specific values of such E are then determined at Step 3 without violating the nominal LQ optimality requirement. However, if this step fails, we must determine Sand E dependently so as to satisfy the small gain condition (19). If, in addition, such Sand E cannot be found, we have to utilize the freedom available in the choice of V before giving up the requirement C2.
Step2) Choose O'i > 0 (1 ~ i ~ m) so as to satisfy the small gain condition (19) on the transfer matrix Ge(s). If no such O'i can be chosen, then go back to Step 1 and rep eat the process until the desired Sand E are obtained. Up to this point , no mention has been made about the choice of the remaining parameter V . As is clear from the foregoing development, this matrix seems to have no essential effect on the two requirements Cl and C2. Therefore we set V = I for simplicity of the design procedure. 4.5. ILQ Robust Servo Design Method Summarizing the foregoing discussion , we obtain 441
5
statement 1) holds iff (13) has a solution P > 0 of the form (14) with A = E- 1 , or equivalently, for some P > 0 of this form, the positive definite matrix
CONCLUSION
By taking the Inverse LQ approach, we have develpoted an optimal servo design method in which quadratic stability can also be taken into consideration in the design process of desired tracking systems. In the course of this development, we clarified the control structure of the tracking system of our interest, i.e., ILQ servo system, as well as the physical meaning of its design parameters in relation to this structure. By making full use of this structure, we were able to obtain not only the simple design algorithm but also the simplified derivation of the existing ILQ design method as a side effect. One important subject for future study along this line is to clarify the role of the parameter V, especially in relation to robust stabilization, together with its effective way of determination. 6
P := TT PT,
V~l]
T := [_IK
of a block diagonal form P = diag{Y, EA} diag{Y, I} by the definition, satisfies
L(P;T- 1 A eT,T- 1 B e, Ke T ) > 0 Substituting this P together with (8c) and (12) into this inequality and rewriting yields (16), or equivalently
D>O and yA+J.\Ty+yB1Fy+cTc<0 The latter inequality is equivalent to the conditions (17b) and (17 c) as was shown in Khargonekar et al.(1990). This completes the proof.
Proof of Proposition 5
REFERENCE
With K given by (32), we first derive (33) by showing that these normalized gains satisfy their definitive equation (22). Substituting (32) and (33) into (22) we see that this is true if
Anderson. B. D. O. and J. B. Moore (1990). Optimal Control: Linear Quadratic Method, Prentice-Hall, New Jersey. Barmish, B. R. (1985). Necessary and Sufficient Conditions for Quadratic Stabilizability of an Uncertain System. J. Optimiz. Theory Appl. 46, 399-408. Douglas, J., and M. Athans (1992) . R0bust Linear Quadratic Designs with Respect to Parameter Uncertainty. Proc. 1992 American Control Conference, 2905-2910. Fujii, T. (1987) . A new approach to LQ design from the viewpoint of the inverse regulator problem. IEEE Trans. Aut. Control AC-32,
N",A + MoC
= N.p,
N",B
or equivalently for each i, 1 ~ i CitPi(A)A+~i(O)Ci
= ci~i(A),
~
=M
m
citPi(A)B
= Mi
holds. The left relation follows immediately by multiplying the left relation of (31) by sand substituting s = A, and finally left-multiplying Ci. The right relation follows by substituting the right relation of (31) and noting the definitions of di and M i . We now remains to show that (32) and the latter part of (33) satisfy (27). Differentiating (lb) and substituting the nominal system equation (la) with F = 0 yields by definition of di :
99~1004 .
Fujii, T., Y. Nishimura, T. Shimomura and S. Kawarabayashi (1987). A Practical Approach to LQ design and its application to engine control. Proc. 10th IFAC World Congress, pp. 253-258, Munich. Gilbert, E. G. (1969). The decoupling of multivariable systems by state feedback. SIAM J. Ctrl. and Opt., 7, 50-63. Khargonekar, P. P., I. R. Petersen and K. Zhou (1990). Robust Stabilization of Uncertain Linear Systems: Quadratic Stabilizability and Hoo Control Theory. IEEE funs. Aut. Control AC-35, 356-361. Neto, A. T., J. M. Dion and L. Dugard (1992) . Robustness Bounds for LQ Regulators. IEEE Trans. Aut. Control AC-37, 1373-1377. Soroka, E., and U. Shaked (1984). On The Robustness of LQ Regulators. IEEE Trans. Aut. Control AC-29, 664-665.
dkYi(t)
dr
= ciAkx(t)
crI'yi(t)
(35)
dtd.
where Yi(t) denotes the i-th component of y(t). Taking the Laplace transform of these equations, multiplying each of them by O'ik (0 ~ k ~ di 1) and finally summing up these equations from k = 0 to k = di yields by definition of ~i(S): ~i(S)Yi(S)
=
Ci~i(A)X(s)
+ MiU(s)
Substituting the control law U(s) = -KX(s) + KJR(s) with K = M- 1 N.p and KJ = M- 1 Mo yields i = 1,"',m
APPENDIX: PROOFS
By definition of Wd(S) as in (26) , this is equivalent to saying that (27) holds as required.
Proof of Proposition 1 As is clear from Lemma 1 and Remark 1, the 442