∇h1 ∇h2 ∇
h3
(xg 3)
G3
xg0
fc0
f˜g1 xg1
fc1
f˜g2 xg2
fc2
f˜
g3
xg3
fc3
Figure 2-3: Visualization of the GPAW scheme. K is the unsaturated region, bounded by hy-persurfaces H1, H2, and G3. Here, xgi := xg(ti), the supporting hyperplane of G3
at xg3 is H3(xg3), and the projection of fci := fc(xg(ti), y(ti), r(ti)) onto Hi yields f˜gi := RI∗(xg(ti), y(ti), r(ti))fc(xg(ti), y(ti), r(ti)). Observe that at each time instant, the combinatorial optimization subproblem (2.31) determines an optimal combination of active constraints to project onto, in particular at the points xg2 and xg3.
2.6 Approximate Nominal Controller
In Section 2.5, restrictions to nominal controllers of the form (2.26) were made to achieve controller state-output consistency. Here, we show that for nominal controllers of general structure (2.24), an arbitrarily close approximating controller can be constructed that has the required structure of (2.26), i.e. with output equation depending only on its state.
Then GPAW compensation can be applied to this approximate controller, yielding the same desirable properties. Note that this construction is not unique, and similar ideas have been discussed in [128, Remark 9].
The main idea is to replace the signal components in the controller output equation that are not part of the controller state by its low-pass filtered signal, and design the low-pass filter such that its bandwidth is much larger than the effective bandwidth of the closed-loop system. It is clear that the approximation will be enhanced as the bandwidth of the low-pass filter is increased. Importantly, the main purpose of this low-low-pass filter is not for noise rejection or performance/robustness enhancements.
Consider the nominal controller
˙xc= fc(xc, y, r), xc(0) = xc0,
uc= gc(xc, y), (2.37)
whose output equation depends not only on the state, but on measurement y as well. For simplicity, we have assumed that the output equation is not dependent on the reference input r. If it indeed does, the treatment is similar.
Remark 2.22. When gcdepends on the measurement y as in (2.37), the closed-loop system
comprising the plant (2.34) and controller (2.37) with u := ucwill contain an algebraic loop
whenever ∂g∂u∂g∂yc 6≡ 0.
Consider augmenting the controller state to be ˜xc := (xc, ˜y), with ˜y = y. Then, by replacing y with ˜y in the controller output equation, we have uc= gc(xc, ˜y) = gc(˜xc), which is the desired form in (2.26). The state equation of the augmented controller with state ˜xc
needs to satisfy
˙xc= fc(xc, y, r), ˙˜y = ˙y. (2.38) Clearly, if the functions f and g in (2.34) are known exactly, realization of (2.38) is straight-forward,17 by taking the time derivative of y in (2.34) and using the knowledge of f and g.
We avoid making such a conservative assumption by using an approximation.
Consider ˜y obtained as the output of an exponentially stable, unity DC gain low-pass filter with input y, parameterized by a∈ (0, ∞)
˙˜y = a(y − ˜y), y(0) = y(0).˜
It can be seen that ˜y(t) → y(t) for all t ≥ 0 as a → ∞, so that the solution of the approximating controller
˙xc= fc(xc, y, r), xc(0) = xc0,
˙˜y = a(y − ˜y), y(0) = y(0),˜ uc= gc(xc, ˜y),
(2.39)
can be made arbitrarily close to the nominal controller (2.37). While this can be shown formally for any fixed y : [0,∞) → Rp and r : [0,∞) → Rnr using singular perturbation theory [37, Chapter 11, pp. 423 – 459], the larger question is the effect of the approximation on the closed-loop system, which we discuss next.
The closed-loop system described by the feedback interconnection of the plant (2.34) and approximate controller (2.39) with u := uc is described by
˙x = f (x, sat(gc(xc, ˜y))),
˙xc= fc(xc, g(x, sat(gc(xc, ˜y))), r),
˙˜y = g(x, sat(gc(xc, ˜y)))− ˜y,
(2.40)
where := a1. Observe that when = 0, we recover the exact closed-loop system obtained with controller (2.37), which corresponds to the reduced system in the singular perturbation framework. System (2.40) is referred as the approximate system when > 0, and the exact system when = 0. When we assume existence and uniqueness of solutions18 to the exact system, then (2.40) is a standard singular perturbation model [37, p. 424]. It can be shown
17When the closed-loop system contains an algebraic loop, there are additional difficulties on well-posedness of the feedback interconnection.
18Recall that in the anti-windup context, the nominal controller has been designed to achieve some desired performance. Existence and uniqueness of solutions to the closed-loop system is usually guaranteed even when not explicitly sought in the control design.
that if g and gc are such that the eigenvalue condition19 [37, p. 433]
Re
λ ∂g
∂u
∂gc
∂y(x, xc)− I
< 0,
holds uniformly for all (x, xc) in some domain, then the origin of the associated boundary layer model for the singular perturbation model (2.40) is exponentially stable. With this, and assuming existence and uniqueness of solutions of the exact system, [37, Theorem 11.1, p. 434] shows that on any finite time interval, the solution of the approximate system can be made arbitrarily close to the solution of the exact system when is sufficiently small (or equivalently, a is sufficiently large). When the equilibrium of the exact system is exponentially stable, [37, Theorem 11.2, pp. 439 – 440] shows that the result extends to infinite intervals.
Observe that for input-constrained LTI systems driven by LTI controllers, local expo-nential stability is usually guaranteed, so that the infinite time approximation result holds.
If the exact system is not exponentially stable and the finite time approximation result in-dicated above is not sufficient, repeating the analysis with the approximate controller may be required. Because the approximation can be made arbitrarily well, it is likely that the approximate controller will be able to achieve the control objectives as well.
Remark 2.23. The approximate controller (2.39) requires the augmentation of the controller state. We note that the (q + m)-th order controller with state (xc, uc) and output uc,
˙xc= fc(xc, y, r), xc(0) = xc0,
˙uc= ∂gc(xc)
∂xc fc(xc, y, r), uc(0) = gc(xc0),
is a non-minimal (equivalent) realization of the nominal controller (2.26). For the preceding augmented controller, it can be shown that GPAW compensation with parameter Γ = I ∈ R(q+m)×(q+m) yields effectively no anti-windup compensation. This suggests that controller state augmentation should always be done with caution, and using a minimal realization [50]
would likely be more appropriate. We leave the study of the implications of controller state
augmentation as future work (see Section 7.1.10).
In summary, for controllers of general structure (2.24), an arbitrarily close approximat-ing controller can be constructed that has the form of (2.26), where the output equation depends only on the controller state. Then GPAW compensation can be applied to the ap-proximate controller yielding controller state-output consistency (see Theorem 2.5.3). The approximate controller constructed in this section will be of higher order than the exact nominal controller. In this sense, application of the GPAW scheme on the approximate controller can be seen to be analogous to the case of employing dynamic anti-windup com-pensators, where additional states are employed. Sections 2.8.1, 6.1.2, and 6.2.3 illustrate how the construction presented can be applied in modified form.
Other than the basic construction presented here and Section 2.5, alternative ways to realize GPAW-compensated controllers are shown in Section 4.1 as well as Appendices A and B. For ease of reference, we summarize the procedure to apply GPAW compensation in Appendix C.
19Here, Re(λ(A)) means real part of all eigenvalues of matrix A, and ∂g∂u∂g∂yc(x, xc) is shorthand for
∂g
∂u(x, sat(gc(xc, ¯h(x, xc))))∂g∂yc(xc, ¯h(x, xc)) where ˜y = ¯h(x, xc) is an isolated real root of the equation g(x, sat(gc(xc, ˜y))) − ˜y = 0.