Data-based uncertainty modeling of MIMO systems

Many robust control design methods require a linear model consisting of nominal model augmented by an uncertainty description. A general form for such a model is

(1)     G = G₀ + H₂₁Δ(I-H₁₁Δ)^-1H₁₂

where G is the transfer function of the true system, G₀ is a nominal model, and Δ is a perturbation causing uncertainty about the true system. Depending on the particular type of uncertainty (additive, input or output multiplicative, inverse types of uncertainty, combinations of various types of uncertainty), H₁₁, H₁₂ and H₂₁ can contain combinations of (known) constant matrices and the nominal model.

Assume that we have information about the true system in the form of a number of possible transfer functions G_k, k = 1,...,N. The nominal model and the perturbation Δ_k associated with G_k are unknown, but they have to satisfy

(2)     G_k = G₀ + H₂₁Δ_k(I-H₁₁Δ_k)^-1H₁₂ ,   k = 1,...,N.

How should G₀ be determined? It has been shown that ||Δ||_∞ is a control relevant measure of the distance between G and G₀ for models of the form (1) and that the achievable stability margin by feedback control is inversely proportional to this distance. For a given type of uncertainty model, this suggests that G₀ should be determined by solving the optimization problem

(3)     min_G0 max_k ||Δ_k||_∞

subject to the appropriate data matching condition (2). Obviously, the type of uncertainty model giving the smallest minimum is the best one according to this measure.

If information about the system is obtain through identification, input-output data are available. An attractive way of removing noise from the output is to fit a model G_k to the data and to calculate a noise-free output y_k by y_k = Gu_k, where u_k is the input in experiment k. Since the purpose of the experiments in this context is to excite the system in various ways, the inputs do not tend to be persistently exciting in all individual experiments. Thus, G_k only applies to the particular input u_k, and the relevant information is input-output data {u_k, y_k}, k = 1,...,N. This means that the model matching condition (2) should be replaced by the input-output matching condition

(4)     y_k = G₀u_k + H₂₁Δ_k(I-H₁₁Δ_k)^-1H₁₂u_k .

It can be shown that the use of (4) instead of (2) results in a less conservative uncertainty model.

Our modeling approach is to model G₀ in the frequency domain using sampled frequency responses of the input-output data. Because of the availability of G_k, these are easy to calculate for standardized inputs. The available information is thus {u_k(jω), y_k(jω), ω∈Ω}, k = 1,...,N, and we solve the optimization problem frequency-by-frequency, i.e.

(5)     min_G0(jω) max_k ||Δ_k(jω)||₂   s.t. (4), ∀ω∈Ω.

The uncertainty Δ_k is assumed to be unstructured.

For some types of uncertainty, the optimization problem can easily be formulated as a convex optimization problem. Additive uncertainty, for example, which in its basic form is described by

(6)     y_k = G₀u_k + Δ_ku_k , k = 1,...,N,

results in the optimization problem

(7)     min_G0γ  s.t. matrix(γI, y_k - G₀u_k; (y_k - G₀u_k)^*, u_k^*u_k) >= 0, k = 1,...,N, ∀ω∈Ω

which is a convex optimization problem. Here A^* denotes the complex conjugate transpose of A and P>=0 denotes that P is positive semidefinite.

Many types of uncertainty descriptions do not readily give a convex optimization problem. For example, an output multiplicative uncertainty described by

(8)     y_k = G₀u_k + Δ_kG₀u_k , k = 1,...,N,

results in the optimization problem

(9)     min_G0γ  s.t. matrix(γI, y_k - G₀u_k; (y_k - G₀u_k)^*, u_k^*G_0^*G₀u_k) >= 0, k = 1,...,N, ∀ω∈Ω

which is non-convex due to the appearance of G_0^*G₀. An iterative solution by keeping G_0^*G₀ fixed during each iteration tends to produce local minima which are non-global. However, we show how this optimization problem, and similar ones for some other types of uncertainty, can be reformulated as a convex optimization problem.

For control design, G₀ is needed as a transfer function or a state-space model. In principle, we should determine such a model by replacing G₀ in the appropriate consistency relations, like those appearing in (7) and (9), by a suitable parameterization of G₀ . However, so far we have not been able to obtain a satisfactory solution in that way. Instead, we have fitted a model to the calculated frequency responses G₀(jω), ω∈Ω. A drawback of this approach is that min||Δ(jω)||₂ will increase, usually also min||Δ||_∞, sometimes even drastically. We have studied various approaches of reducing the effects of this drawback.