In this section the H∞-methodology is used to translate the control goals presented in Section 4.1 into a mathematical optimization problem, from which a robust controller results.
4.2.1 Nominal Stability
The stability of the multivariable feedback control system shown in Figure 4.1 is determined by the generalized Nyquist stability criterion [16].
Theorem 1 (Generalized (MIMO) Nyquist theorem) Let Pol denote the number of open-loop unstable poles in P(s)C(s). The closed-open-loop system with open-loop transfer function P(s)C(s) and negative feedback is stable if and only if the Nyquist plot of det(I + P(s)C(s))
• makes Polanti-clockwise encirclements of the origin, and
• does not pass through the origin
The Generalized Nyquist Stability Criterion will be used in assessing not only nominal stability but also robust stability of an uncertain closed loop system, see Section 4.2.3.
30 4.2. ROBUST CONTROL DESIGN BASED ON H∞ THEORY
4.2.2 Nominal Performance
A closed-loop system achieves nominal performance if the performance objective is satisfied for the nominal plant P0(s). The performance of a vibration isolation system is determined by the level of disturbance rejection. Disturbances, entering at the plant output, consist of disturbances directly acting on the isolated mass, such as exciting force generated in a mounted operating machine, airflow from the air conditioning and the sound pressure on one hand and vibrations of the chassis transmitted to the payload on the other hand. From Figure 4.1 it is easily seen that the sensitivity function,
S(s) = (I + P (s)C(s))−1, (4.5)
is the transfer function from disturbance d(s) to output y(s). This means that the nominal performance of the AVIS is specified by the sensitivity function. For good disturbance rejection, that is for d(s) to affect y(s) to the least extent, the sensitivity function should be small.
For scalar systems, the absolute value of the complex valued frequency response |S(s)| is often used as a measure of the size of a transfer function S(s). However, for multivariable systems the frequency response will be a complex valued matrix. Therefore a scalar measure of the size of a complex valued matrix is needed. It has become a standard practice to use the H∞-norm as gain measure.
The H∞-norms of SISO and MIMO system are defined as follows, [5]:
• Let H(s) be a transfer function of a stable SISO system with frequency response ˆh(ω). The H∞ norm of H(s), denoted by kH(s)k∞is defined as
kH(s)k∞:= max
ω∈R|ˆh(ω)|. (4.6)
• Let H(s) be a stable multivariable transfer function. The H∞ norm of H(s), is defined as kH(s)k∞:= sup
ω∈R
σ(H(jω)). (4.7)
where σ denotes the maximum singular value.
Often a performance specification for robust control is given as a weighted sensitivity specification,
kWy(s)S(s)Vd(s)k∞≤ 1, (4.8)
where Vd(s) and Wy(s) denote the input and output weight respectively. The weighting matrices are usually frequency dependent and typically selected such that weighted signals ˜d and ˜y are of magnitude 1, that is, the norm of the transfer function from ˜d to ˜y should be less than 1. The feedback control scheme including weights, casted in the standard H∞/µ-framework, is shown in Figure 4.3.
4.2.3 Robust Stability
A closed-loop system achieves robust stability if the closed-loop system is internally stable for all perturbed plants about the nominal model up to the worst-case model uncertainty.
G C
M
u vK
P ( s )
++
d y y~
d~ V d - W y
w~
z~
Figure 4.3: General control configuration for nominal performance
Additive Uncertainty
In order to assess robust stability, first the model uncertainty has to be defined. Because the H∞ theory is based on the assumption of norm bounded, unstructured complex perturbations, only this type of model uncertainty is considered at this point. The simplest way to represent the discrepancy between the model and the true system is by taking an additive uncertainty model:
∆(s) = Pt(s) − P0(s) (4.9)
where Ptrepresents the true plant, P0 denotes the nominal model and ∆(s) obeys:
k∆(jω)k∞≤ γu(ω), ∀ω ≥ 0 (4.10)
The perturbation ∆(s) is thus a full, norm-bounded complex matrix. In general, it is convenient to normalize the uncertainty using weights, i.e., represent the uncertainty model by
∆(s) = W∆(s) ˜∆(s)V∆(s), (4.11)
where W∆(s) and V∆(s) denote the output and input weight respectively, such that
k ˜∆k∞< 1. (4.12)
Thus, it can be concluded that the weighted model uncertainty, ˜∆, belongs to the following norm-bounded subset
∆u= { ˜∆ ∈ Cn×n| k ˜∆k∞< 1}, (4.13) where n is the order of the plant. The feedback control configuration with additive uncertainty is depicted in Figure 4.4(a).
Small Gain Theorem
The main results with respect to robust stabilization of dynamical systems follow straightforwardly from the Small Gain Theorem. Consider again the feedback control configuration with additive uncertainty depicted in Figure 4.4(a). To study the stability properties, the system of Figure 4.4(a) can also be presented in the H∞/µ-framework as shown in Figure 4.4(b). It can easily be seen that
M (s) = Wu(s)R(s)V∆, (4.14)
where R(s) = C(s) (I + P (s)C(s))−1 is the control sensitivity. The stability properties of the configuration of Figure 4.4(b) are determined by the Small Gain Theorem, [5].
32 4.2. ROBUST CONTROL DESIGN BASED ON H∞ THEORY
C Pu
e y
- D
V
uW D~
+ +
u~
D
D~
(a)
M ( s ) D~ u~
D~
(b)
Figure 4.4: Feedback configuration including additive uncertainty
Theorem 2 (Small Gain Theorem) Suppose that the systems M and ∆ are both stable. Then the autonomous system determined by the feedback interconnection of Figure 4.4(b) is asymptoti-cally stable if the spectral radius
ρ(M (jω)∆(jω)) < 1, ∀ω ∈ R, ∀∆ ∈ ∆, (4.15) where
ρ(M ∆) := max
i |λi(M ∆)|. (4.16)
The Small Gain Theorem can now be used to asses the closed loop stability under unstructured, norm-bounded perturbations like (4.13). In [15] it is proved that
ρ(M (jω)∆(jω)) < 1, ∀ω ∈ R, ∀∆ ∈ ∆u
⇔ kM (s)k∞≤ 1 (4.17)
Summarizing, the robust stability objective of the additively perturbed closed loop system in Fig-ure 4.4(a), is to design a nominally stabilizing controller C(s) such that kWuRV∆k∞is minimized.
Robust stability is achieved if
kWu(s)R(s)V∆(s)k∞≤ 1. (4.18)
4.2.4 Robust Performance
A closed-loop system achieves robust performance if the closed-loop system is internally stable for all perturbed plants about the nominal model up to the worst-case model uncertainty and the performance objective is satisfied for the perturbed plant.
The robust performance objective is derived from (4.8) with nominal sensitivity function S0(s) replaced by the perturbed sensitivity function St(s):
kWy(s)St(s)Vd(s)k∞≤ 1, (4.19)
Consider the feedback control configuration including model uncertainty and weights shown in Figure 4.5(a), which is equivalent to the control configuration of Figure 4.5(b), with
N (s) =
−WuRV∆ −WuRVd
WySV∆ WySVd
, (4.20)
where R and S are the control sensitivity and sensitivity respectively.
C P + +u d
e y
- y~
d~
D
V
u
W
dV
yW
D~
+ +
u~
D
D~
(a)
N
d~ y~u~
D~
D~
(b)
Figure 4.5: Feedback configuration for robust performance
N
d~ D~
u~
y~a
D~
Figure 4.6: N∆-structure
If the perturbation structure is augmented with a full complex performance block ˜∆p(s) with k∆pk∞< 1, such that
∆˜a(s) =
∆(s)˜ 0 0 ∆˜p(s)
(4.21) it can be noticed that the robust performance condition, (4.19), is similar to the robust stability condition, (4.17), see also Figure 4.6.
Thus the Small Gain Theorem, (4.15), can now be used to asses the robust performance under unstructured norm bounded perturbations. The robust performance objective of the additively perturbed closed loop system, which is shown in Figure 4.5(a), is to design a nominally stabilizing controller K(s) such that kN (K(s))k∞, is minimized. Robust performance is achieved if
kN (s)k∞≤ 1. (4.22)
4.2.5 Computation of the H
∞Controller
From Section 4.2.4 it appears that the H∞ optimal control problem is to find all stabilizing controllers K(s) which minimize kN (K(s))k∞. In practice, it is usually not necessary to obtain an optimal controller for the problem and it is often computationally and theoretically simpler to design a suboptimal one.
Let γmin be the minimum value of kN (K(s))k∞ over all stabilizing controllers K(s). Then the H∞suboptimal control problem is, given a γ > γmin, to find all stabilizing controllers K(s) such that
kN (K(s))k∞< γ. (4.23)
This can be solved efficiently using the algorithm of [7] and by reducing γ iteratively, an optimal solution is approached. However, it took almost a decade of research to find this so called state-space solution of the H∞ problem. The solution is described in [7] and involves only two Ricatti equations, yielding a controller with the same order as the generalized plant.