In this section we discuss robust stability and performance. We relate robust stability to the input-output properties of the system using the small gain theorem. We then intro-
Figure 2.2: The standard robust control block diagram for a model subject to uncertainty. Generally a norm bound on ∆ specifies the amount of uncertainty that a model can have before a desired property is lost (e.g., if the model is stable fork∆k ≤1, this implies robust stability).
duce the concepts of non-normal linear operators and pseudospectra. Finally we relate the pseudospectra of an operator to robust stability of that operator.
In order to study a physical system, usually one constructs a simple model that describes the evolution of the system as accurately as possible. Models are by definition idealized versions of the actual physical process and as such there is some uncertainty inherent in any model. These uncertainties may represent physical phenomena that are difficult to characterize, errors in model parameters, conditions that can be characterized by adding additional complexity to the model, or any other unmodeled conditions that tend to be present in experiments or numerical simulations.
In order to understand how an uncertainty or modeling errors affect the behavior of a model one can study its “robustness” [15, 17]. A model/property is said to be robust to a particular uncertainty/disturbance if it maintains that property in the face of the uncertainty/disturbance. Two common characteristics that are often discussed in terms of robustness are stability and performance. One generally illustrates this idea through the diagram shown in Figure 2.2 [15] where the block labeled model may represent, for example, the transfer functionHand the ∆ represents any uncertainties/disturbances that influence the system.
2.5.1 Robust Input-Output Response
In order to make the concept of robust stability concrete it is useful to study system (2.2) in terms of its resolvent. If a system satisfies the spectrum determined growth condition
(2.6), then exponential stability of the generator of Ais equivalent to
sup Re(z)>0kRz
(A)k<∞.
For a system of the form (2.1a) with C = I, B = I and input d, the resolvent can be determined as the transfer function fromdto the stateϕ, (we denote this byT to distinguish it from the more general case whereB,C 6=I). Then in the transformed space the maximum modulus principle [15] means that exponential stability reduces to a condition on the infinity norm H∞of this special case of the system (2.1a).
Using this idea, the problem of robust stability can be precisely formulated as stability of a system
∂ϕ
∂t = (A+ ∆)ϕ (2.23)
with ∆ accounting for model uncertainty. With this simple “unstructured uncertainty”, the small gain theorem provides a test for robust stability.
Theorem 2.5.1. [15] If both ∆and T are stable, then the feedback interconnection shown in Figure 2.2 is stable if k∆TkL2 <1.
The small gain theorem provides a characterization of robust stability in terms of the system gain, which is closely related to the resolvent of the operatorAin (2.1). In the next section we take a closer look at Rz(A) for particular regular values z ∈ C, the so-called pseudospectra. We use this idea to illustrate a slightly different version of the small gain theorem.
2.5.2 Relationship with Pseudospectra
The pseudospectra ofAare closely related to its spectra. Recall that the spectrum ofAcon- sists of the pointsλ∈Csuch that (λI−A)−1 does not exist (as a bounded linear operator).
This definition can also be interpreted to mean that perturbations with a frequencyλcan have unbounded amplification [88, 90], which can roughly be stated as k(λI−A)−1k=∞. In many applications one might also be interested in regular pointszthat produce large or small resolvent.
Definition 2.5.1. [89] Let Abe a closed linear operator on a Banach spaceB andε >0 be
Figure 2.3: Block diagram of the system (2.25) that can be represented by (2.1) for appli- cation of a small gain theorem.
equivalent conditions: Λε(A) = © λε∈C: ° °(λεI − A)−1 ° °≥ε−1 ª , (2.24a)
λε∈σ(A) or k(λεI − A)uk< ε for some u∈ D(A) withkuk= 1, (2.24b)
where D(A) is the domain of A. If k(λεI − A)uk < ε as in (2.24b), then λε is an ε-
pseudoeigenvalue of A and u is the corresponding ε-pseudoeigenfunction (or pseu- domode).
Clearly Λε(A) is a superset of the eigenvalues of A, which can be thought of as the 0- pseudospectra Λ0(A). An equivalent definition of theε-pseudospectra that is also useful in the context of robust control theory is the following.
Definition 2.5.2. [89] The ε-pseudospectrum of A is the set of complex numbers λε ∈C
such thatλε∈λ(A+E) for some bounded linear operator E withkEk< ε andε >0. i.e.,
the eigenvalues of some nearby linear system defined by A+E.
This is directly related to determining the stability of (2.23) and Theorem 2.5.1 with
E representing ∆. In order to emphasize the connection we look at a more structured dynamical system:
∂ϕ
∂t = (A+B∆C)ϕ . (2.25)
This can be represented by the system (2.1) if we defined:= ∆φas shown in Figure 2.3. Then robust stability of (2.25) means that
sup Re(z)kRz
where Rz(A+B∆C) is the resolvent of A+B∆C. The expression (2.26) precisely defines
E = ∆ in Definition 2.5.2 when B=C=I. The small gain theorem for the system (2.25) can then be stated.
Theorem 2.5.2. The system in Figure 2.3 is robustly stable for all k∆kL2 if and only if
its spatio-temporal response,H satisfies kHkL2 <
1 ε.
Equation 2.24a in Definition 2.24 shows that the points that do not satisfy Theorem 2.5.2 (i.e., µ∈C such that kHkL
2 ≥
1
ε) are essentially the ones inside the ε-pseudospectrum of
A.
The final point that we discuss in this chapter has to do with the pseudospectra of non-normal linear operators.
Definition 2.5.3. A linear operatorAon a Hilbert spaceHis said to be normalifAA∗=
A∗A, where A∗ is an appropriately defined adjoint.
A non-normal linear operator is one that is not normal.
For the finite-dimensional matrix case, normality is equivalent to the matrix A having a complete set of orthogonal eigenvectors.
The ε-pseudospectra of a normal linear operator are the points z ∈ C that measure a
distance (in the appropriate metric) of at most εfrom the spectra. This is not the case for a non-normal operator, in which case theε-pseudospectra may be much further away. This means that systems governed by non-normal operators can experience large amplification
(pseudoresonance) at frequencies that are far from the resonant points (i.e., the spectra) [88].
The properties of non-normal linear operators are of interest because wall-bounded shear flows are generally governed by such operators. In the next chapter, we will discuss these flows in more detail and specifically discuss how non-normality is related to important flow phenomena.