2.4 Finite-spectrum Assignment (FSA)
The SP-based control scheme is very effective for control of stable time-delay systems. While the modified Smith predictor was being developed for unstable systems, another effective control strategy, known as finite-spectrum assign- ment nowadays, was developed for unstable systems as well. This strategy can address delays not only in the input/output channel, but also in the states.
The delays can be multiple, commensurate and even distributed. Here, only the case with a single input delay is discussed, as the book focuses on systems with a single input/output delay. See [70, 108, 109, 146, 148, 149] for other cases and [49, 144] for a different version using the pole-assignment argument.
Consider a system described in the state-space realization as x˙(t) =Ax(t) +Bu(t−h),
y(t) =Cx(t). Then, the plant transfer function is
G(s) =P(s)e−sh= A B
C 0
e−sh. The FSA adopts a state feedback control law
u(t) =F xp(t) using the predicted statexp(t)given by
xp(t) =eAhx(t) +
h
0 eAζBu(t−ζ)dζ.
The resulting closed-loop system, having a finite spectrum, is stable ifA+BF is stable. Similar to the predictor-based control schemes, the delay term is removed from the design process.
This strategy will be discussed further with comparison to the modified Smith predictor in the next section. Simulation examples will be given in Part II with different implementations of the distributed delay
v(t) =
h
0 eAζBu(t−ζ)dζ in the control law.
2.5 Connection Between MSP and FSA
2.5.1 All Stabilising Controllers for Delay Systems
Assume that the control plant isP(s)e−sh, of which the delay-free partP has the following realization:
P = A B
C 0
,
where (A, B) is stabilisable and (C, A) is detectable. LetF and L be such thatA+BF andA+LCare Hurwitz. Then all stabilising controllers [90, 91]
incorporating a modified Smith predictor Z =Ce−Ah(I−e−(sI−A)h)ã
A B I 0
, (2.12)
can be parameterised as shown in Figure 2.17, where J(s) =
⎡
⎣A+BF+eAhLCe−Ah −eAhL B
F 0 I
−Ce−Ah I 0
⎤
⎦
and Q(s) is arbitrary but stable. This controller parameterisation involves one-degree-of-freedom. Another parameterisation involving two-degrees-of- freedom can be found in Chapter 9.
y
) (s J
e hs
s P ( ) −
) (s Q
u
Z η
ε
yp
Figure 2.17. Stabilising controllers for processes with dead time
2.5.2 Predictor–Observer Representation: MSP Denote the state vector of J(s)byxJ, then
u=F xJ+η, ε=−Ce−AhxJ+yp. The state equation ofJ(s)is given by
x˙J = (A+BF+eAhLCe−Ah)xJ−eAhLyp+Bη, or equivalently, by
e−Ahx˙J = (A+LC)e−AhxJ−Lyp+e−AhB(F xJ+η)
= (A+LC)e−AhxJ−Lyp+e−AhBu.
2.5 Connection Between MSP and FSA 41 Using the above formulae, all the stabilising controllers shown in Figure 2.17 can be represented as shown in Figure 2.18. It consists of an output predictor Z, a state observer and a state feedback. The state observer actually observes the states of the delay-free systemA B
I 0
because, in the nominal case,
xJ =
A+LC e−AhB eAh 0
u−
A+LC L eAh 0
(Z+P e−sh)u
=
A+LC e−AhB eAh 0
u−
A+LC L eAh 0
A B
Ce−Ah 0
u
=
A+LC I eAh 0
A I
−LC I
e−AhBu
=
⎡
⎣A+LC−LC I
0 A I
eAh 0 0
⎤
⎦e−AhBu
= A B
I 0
u.
2.5.3 Observer–Predictor Representation: FSA
As mentioned in Section 2.4, the finite-spectrum assignment for time-delay systems is a state feedback control law
u(t) =F xp(t) using the predicted state
xp(t) =eAhx(t) +
h
0 eAζBu(t−ζ)dζ.
If the statex(t)is not available for prediction, then a Luenberger observer is needed to re-construct the state from the output y and the control u. It is easy to check that, in the nominal case,
xo=
A+LC I I 0
ã(Be−shu−Ly)
=
A+LC I I 0
ã
A I
−LC I
Be−shu
= A B
I 0
e−shãu
gives the observed statexoof the plant. Using these formulae, the FSA control structure (using output feedback via y) can then be depicted in Figure 2.19, assuming temporarily Q(s) = 0. It consists of a state observer and a state
-
e hs
s P( ) −
)1
(sI−A−LC − L
e−Ah
eAh
F
) (s Q
C
u y
ε
- xJ
η
B
Z
Observer
Predictor
yp
Figure 2.18.Predictor–Observer Representation: MSP scheme
xo
- xp
e hs
s P( ) −
Z
) 1
(sI−A−LC − L
e−hs
eAh
F
) (s Q
C u y
ε
-
η
B Predictor
Observer
v
x
Figure 2.19. Observer–Predictor Representation: FSA scheme
2.5 Connection Between MSP and FSA 43 predictor. As a matter of fact, this is exactly the central controller given in Figure 2.17, as described in [90, 91]. It is also easy to check that
ε=−Cxo+y.
Hence, all the stabilising controller given in Figure 2.17 can also be represented as the observer-predictor structure shown in Figure 2.19, where the distributed delay
Zx(s) = (I−e−(sI−A)h)ã A B
I 0
is the transfer function fromutov of the block characterised by the integral v(t) =
h
0 eAζBu(t−ζ)dζ.
2.5.4 Some Remarks
Although the FSA scheme can deal with much more general time-delay sys- tems, e.g., systems with delays in the state and multiple delays, the FSA scheme and the MSP scheme are two equivalent representations of stabilising controllers for systems with a single input delay [91]. The two structures de- picted in Figures 2.18 and 2.19 clearly show the similarities between them.
Roughly speaking, only the order of the predictor and the observer is ex- changed: in the FSA scheme, the observer goes first and then the predictor but, in the MSP scheme, the predictor goes first and then the observer. The only change from the FSA scheme to the MSP scheme is to change/moveZx
toZ ande−shin the observer to e−Ah. Some more insightful observations are:
(i) the predictor in the MSP scheme is anoutput predictor while that in the FSA scheme is astate predictor;
(ii) the observer in the MSP scheme is a state observer of thedelay-free system while the one in the FSA scheme is a state observer of thedelay system (and hence a state predictor has to be used before using the state feedback);
(iii) the free parameterQ(s)may be used to improve the robustness of the FSA scheme w.r.t. the implementation error ofZx;
(iv) in both cases, the central controller is essentially a state feedback controller, using the predicted state (xp in the FSA scheme) or the observed state of the delay-free system (xJ in the MSP scheme).
3
Preliminaries
In this chapter, some preliminaries are collected for later use. These include two important FIR operators which map a rational transfer matrix into FIR blocks, the state-space operations of systems, the chain-scattering approach, algebraic Riccati equations, an important matrix called theΣmatrix, and the L2[0, h]-induced norm.