Three region of attraction (ROA) comparison results were presented in Section 4.4. Of note is Theorem 4.4.3, which yields a prescribed GPAW parameter Γ = ΓT > 0 ∈ Rq×q such that GPAW compensation with parameter Γ ensures “global asymptotic stability” for the equilibrium of the GPAW-compensated system Σg, in the sense that the ROA contains the unsaturated region as in [103, 104]. In this section, we use Theorem 4.4.3 to derive a stability result for the GPAW-compensated system, for which the GPAW parameter is found by solving a linear matrix inequality (LMI) problem [36, Section 2.2.1, p. 9]. As will be explained in Remark 5.1 below, this result applies only to systems with open-loop stable plants. A significant aspect is that the result depends only on properties of the unconstrained system Σu.
Theorem 5.2.1(Global Asymptotic Stability for LTI GPAW-Compensated System). Con-sider the unconstrained systemΣu (5.4). If there exist symmetric positive definite matrices P1= P1T > 0∈ Rn×n and P2 = P2T> 0∈ Rq×q such that
then GPAW compensation with parameter Γ = P2−1 yields systems Σg (5.8) and Σgu (5.9) whose ROAs contain the unsaturated region Rn×K, where K = {¯x ∈ Rq | sat(Ccx) = C¯ cx¯}.
Proof. We will be applying Theorem 4.4.3 by comparing against the ROA of the uncon-strained system Σu (see Remark 4.10). Define P := h
P1 0 0 P2
i
and let V (z) := zTP z be a Lyapunov function candidate for system Σu. Since P1 and P2 are symmetric positive def-inite, P is also symmetric positive defdef-inite, so that V is a positive definite function. It is also clear that V is radially unbounded. Since Au = A BCc
BcC Ac+BcDCc and (5.10) holds, we
have ∂V (¯z)
∂z fu(¯z) = ¯zT(P Au+ ATuP )¯z < 0, ∀¯z 6= 0. (5.11) By [37, Theorem 4.2, p. 124], the origin zeq is a globally asymptotically stable equilibrium for the unconstrained system Σu, so that the ROA of system Σu contains the unsaturated region Rn× K. Since the vector fields fu (5.4) and fgu (5.9) coincide in the interior of the unsaturated region Rn× (K \ ∂K), it follows that zeqis also a locally asymptotically stable equilibrium for system Σgu.5
It can be verified that zeq satisfies (4.29), i.e. it is an equilibrium for the unconstrained system Σuthat lies within the interior of the unsaturated region. Moreover, since the output equation of the nominal controller (5.2) is linear in the controller state, the unsaturated region K is convex (see Remark 4.4), so that K and Rn× K are star domains with kernels ker(K) and Rn × ker(K) respectively (see Remark 4.2 and Corollary 4.2.2). It can be verified that zeq ∈ Rn× ker(K), and that V (z) = V (x, xc) has the form of (4.35) with Vx(x) = xTP1x and Pc = P2. Moreover, since V is continuously differentiable and fu is continuous, it follows that ˙V (z) = ∂V (z)∂z fu(z) is continuous, and is also negative definite
5The vector fields fn and fg also coincide with fu in the interior of the unsaturated region, and zeq is also a locally asymptotically stable equilibrium for systems Σn and Σg. However, these are not needed in the proof.
due to (5.11). By [37, Lemma 4.3, p. 145], there exists a class K function α such that V and fu satisfy (4.33).
All hypotheses of Theorem 4.4.3 are satisfied, and its application shows that GPAW compensation with parameter Γ = Pc−1 = P2−1 yields system Σgu whose ROA contains the unsaturated region Rn× K. From the definitions of fg (5.8) and fgu (5.9), it can be seen that fg and fgu coincide in the unsaturated region, which is a positively invariant set of systems Σg and Σgudue to Theorem 2.5.3. These imply that the ROAs of systems Σg and Σgu within the unsaturated region coincide, and yields the second conclusion, namely that the ROA of system Σg (with parameter Γ = P2−1) also contains the unsaturated region. Remark 5.1. When the matrix operations on the left-hand-side of (5.10) are carried out, it becomes
P1A + ATP1 P1BCc+ CTBcTP2
(P1BCc+ CTBcTP2)T P2(Ac+ BcDCc) + (Ac+ BcDCc)TP2
< 0. (5.12) As implied by [124, Theorem 7.7.6, p. 472], necessary conditions for (5.12) (and (5.10)) to hold are that the diagonal blocks, i.e. P1A+ATP1and P2(Ac+BcDCc)+(Ac+BcDCc)TP2, must be negative definite. Hence it is necessary for A and Ac+ BcDCcto be Hurwitz. This observation actually follows from [182, Proposition 3.5]. This means that Theorem 5.2.1 can only be applied to systems with stable open-loop plants. Moreover, if D = 0, Ac+BcDCc= Ac being Hurwitz means the nominal controller must also be stable. The following example adapted from [182, Example 3.6] shows that A and Ac+ BcDCc being Hurwitz is not sufficient to ensure the existence of P1 = P1T > 0 and P2 = P2T > 0 satisfying (5.10).
Example 5.2.1. Let
Au =
A BCc
BcC Ac+ BcDCc
=−1 2 2 −1
.
It is clear that A = −1 and Ac+ BcDCc = −1 are both Hurwitz. However, it can be verified that Au is not Hurwitz, with eigenvalues of −3 and +1. Hence no P1 = P1T > 0
and P2 = P2T> 0 exist to satisfy (5.10). 4
Necessary and sufficient conditions to ensure existence of P1= P1T > 0 and P2 = P2T > 0 that satisfy (5.10) are available in [182, Theorem 3.10] for a more general case. We note that condition (5.12) (and hence (5.10)) is an LMI, which admits efficient numerical solutions.
This will be demonstrated in the next section on a simple example.
When the LMI (5.10) is feasible, Theorem 5.2.1 yields a GPAW parameter defined by Γ = P2−1. As discussed in Remark 2.25, it is desirable for Γ to have a small condition number [124, p. 336]. In view of this, we formulate a generalized eigenvalue problem [36, Section 2.2.3, pp. 10 – 11] to minimize the condition number of the resultant GPAW param-eter, applicable whenever (5.10) is feasible. Since Γ = P2−1, the definition of its condition number κ(Γ) [124, p. 336] yields
κ(Γ) =kΓ−1kkΓk = kP2kkP2−1k = κ(P2).
Hence minimizing κ(Γ) is equivalent to minimizing κ(P2). From [36, Section 3.2, p. 38], it
can be seen that the solution P2 = P2T> 0 to the generalized eigenvalue problem
P1min,P2,µ,γγ,
subject to P1 > 0, µ > 0, µI < P2< γµI,
P1A + ATP1 P1BCc+ CTBcTP2
(P1BCc+ CTBcTP2)T P2(Ac+ BcDCc) + (Ac+ BcDCc)TP2
< 0, (5.13)
is of minimal condition number, with P1, P2satisfying (5.10). Application of Theorem 5.2.1 remains unchanged, as will be shown in the next section.
5.2.1 Numerical Example
Here, we demonstrate an application of Theorem 5.2.1 on a simple system comprising a sat-urated second-order SISO LTI plant driven by a nominal second-order SISO LTI controller, where the objective is to regulate the system state about the origin.
The unconstrained stable plant is represented by the transfer function G(s) = s2+s+11 , which induces a saturated plant with state-space representation (5.1) where
A = 0 1
−1 −1
, B =0 1
, C =1 0 , D = 0.
Assume a nominal controller with transfer function K(s) =−s222.8s+11+8.6s+25 has been designed to improve the transient response of the system, that is to be interconnected with the plant by positive feedback. The nominal controller has a state-space representation (5.2) where
Ac=−8.6 −6.25
4 0
, Bc=4 0
, Cc=−5.7 −0.6875 .
The matrices (A, B, C, D, Ac, Bc, Cc) define the unconstrained system Σu (5.4) com-pletely, which is all the data required to apply Theorem 5.2.1. Using the LMI solver (or function) feasp of the MATLAB R Robust Control Toolbox [183], symmetric positive defi-nite matrices P1f := P1 and P2f := P2 that satisfy the LMI (5.12) (and hence (5.10)) were found to be
P1f = 1.6090 0.31710 0.31710 0.66628
, P2f = 1.3255 0.13730 0.13730 1.7704
. Thus Theorem 5.2.1 shows that GPAW compensation with parameter
Γf = P2f−1=
0.76056 −0.058984
−0.058984 0.56941
,
yields the GPAW-compensated systems Σg (5.8) and Σgu (5.9) whose ROAs contain the unsaturated region.
Using the solver (or function) gevp of the MATLAB R Robust Control Toolbox [183], the solution to the generalized eigenvalue problem (5.13) was found to be
µ = 1.5046× 10−26, γ = 1.2012,
P1o = 1.9132 0.39208
Since we can always scale the GPAW parameter by a constant positive scalar (see Re-mark 2.15), we can use
Γo = 2.3948× 10−25P2o−1=15.451 −1
−1 13.715
,
as the normalized GPAW parameter. It can be verified that the condition numbers of Γf and Γo are κ(Γf) = 1.4064 > κ(Γo) = 1.1997, which shows a marginal improvement (decrease) when solving the generalized eigenvalue problem (5.13) to obtain P1 and P2.
Remark 5.2. When the nominal controller (with transfer function K(s) = −s222.8s+11+8.6s+25) is represented (equivalently) by matrices
in (5.2), the numerical solutions change significantly. In particular, the condition numbers become κ(Γf) = 22.324 and κ(Γo) = 17.441, representing a drastic deterioration. This shows that the numerical solutions are sensitive to coordinate transformations. Moreover, attempts to use nominal controllers with higher bandwidth, e.g. with K(s) =−s2+9.6s+30.631s+17.4 , have failed in the sense that (5.10) and (5.13) becomes numerically infeasible. This suggests
that Theorem 5.2.1 is a conservative result.
The GPAW-compensated controller (5.5) can be implemented in a few ways summa-rized in Appendix C. Here, we use the closed-form expressions (A.7) in Appendix A for a more efficient solution. The closed-form expressions (A.7) for the GPAW-compensated controller (5.5) with parameter Γ are6
˙xg = Two sets of solutions for the unconstrained system Σu (5.4), nominal system Σn (5.3), and GPAW-compensated system Σg (5.8) are shown in Fig. 5-1. The GPAW-compensated systems with parameters Γf and Γoare denoted by Σgf and Σgorespectively. In Fig. 5-1(a), the plant initial condition is x(0) = (1, 1), while in Fig. 5-1(b), the plant initial condition is x(0) = (2, 2). In both cases, the controller states are set to zero and the plant state is decomposed as x = [x1, x2]T. While Theorem 5.2.1 ensures global asymptotic stability for the origin of Σgf and Σgo in the sense of [103], the time responses in Fig. 5-1 suggests
6Note that cc in (A.7) is given by cc= CcT. Moreover, ∧ denotes the logical AND operator.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Figure 5-1: Comparison of time responses of unconstrained system Σu, nominal system Σn, and GPAW-compensated systems Σgf (Γ = Γf) and Σgo (Γ = Γo), all with zero initial conditions for the associated controllers.
there is little performance improvement when adopting the GPAW scheme. This can be attributed to the conservativeness of Theorem 5.2.1.