On data-driven control: informativity of noisy input-output data with cross-covariance bounds

In this paper we develop new data informativity based controller synthesis methods that extend existing frameworks in two relevant directions: a more general noise characterization in terms of cross-covariance bounds and informativity conditions for control based on input-output data. Previous works have derived necessary and sufficient informativity conditions for noisy input-state data with quadratic noise bounds via an S-procedure. Although these bounds do not capture cross-covariance bounds in general, we show that the S-procedure is still applicable for obtaining non-conservative conditions on the data. Informativity-conditions for stability, $\mathcal{H}_\infty$ and $\mathcal{H}_2$ control are developed, which are sufficient for input-output data and also necessary for input-state data. Simulation experiments illustrate that cross-covariance bounds can be less conservative for informativity, compared to norm bounds typically employed in the literature.

distributed controller design [7] and dissipativity analysis [8], [9] from data, and represent, for example, magnitude, energy and variance bounds on the noise.
In this paper, we consider the problem of determining informativity of input-output and input-state data for control with prior knowledge of the noise in the form of a sample cross-covariance type bound with respect to a user-chosen instrumental signal. Bounds on the sample cross-covariance were introduced in [10] as an alternative to magnitude bounds in parameter bounding identification, given its overly conservative noise characterization, cf. [11] for a comparison of instantaneous and (quadratic) energy type bounds for datadriven control. Our approach to data-driven control extends existing frameworks in two relevant directions: a more general noise characterization in terms of cross-covariance bounds with practical relevance and informativity conditions for control based on input-output data. We provide sufficient conditions for informativity for stabilization, H ∞ and H 2 control, which are also necessary for input-state data.

A. Cross-covariance noise bounds
Consider the sample cross-covariance with respect to the noise e ∈ R p and a variable r ∈ R M , given by The variable r is instrumental and can be specified by the user (as discussed at the end of this subsection), i.e., it is a given variable in the upcoming analysis. We assume prior knowledge on the noise of the form where H u is an upper-bound on the squared sample cross- In generalized form, we write with Q 22 ≺ 0. For Q 11 = N H u , Q 12 = 0 and Q 22 = −I, the bound (5) is recovered. Note that (6) can be rewritten as the bound in [4, [4]. This means that the data informativity results of [4] cannot be used to establish data informativity for general cross-covariance noise bounds. However, the matrix S-lemma in [4,Theorem 13] can still be exploited to obtain necessary conditions for informativity of input-state data with crosscovariance bounds, as shown in Proposition 1 of this paper.
In the state-space representation (3), the matrices A z and B z contain unknown parametersĀ,B and B 0 . Write and so that Λ e and B e are unknown parameter matrices, concatenated with zero rows, and J 1 and J 2 are binary matrices. The set of all pairs (Λ e , B e ) that are compatible with the data is and consider Proof: The first statement follows from (4), (6), and the definition of Σ RQ (U,Y ) . Full-column rank of R − [Z ⊤ − U ⊤ − ] and (9) imply that the last ml + p(l − 1) rows of [Λ e B e ] = col(M 1 , 0) are zero and We denote all (A z , B z ) that are compatible with the data bȳ We have provided a parametrization of (a superset of) Σ RQ (U,Y ) based on the data equation (4). One can equivalently parametrizeΣ RQ (U,Y ) on the basis of the state data equation This leads to an equal setΣ RQ (U,Y ) , but the 'repeated' data in the parametrization contained in Z + := ζ(1) · · · ζ(N ) , would render the evaluation numerically sensitive. Design methods in [4], [6] can yield a parametrizationΣ RQ (U,Y ) based on this state data equation, but with limited applicability to cross-covariance bounds (6), i.e., only if the dimension of r satisfies M ≥ N .
Existing guidelines [10] recommend choosing an instrumental variable r that is correlated with the input u, but uncorrelated with the noise e. Hence, this suggests the choice of (filtered/delayed versions) of the input for r in an open-loop case for data collection, and an external reference/dithering signal for r in a closed-loop case. Moreover, Lemma 1 provides an additional guideline for the choice of r to reduce conservatism in the case of input-output data, has full column rank only if the number of instrumental variables M satisfies M ≥ pl + m(l + 1).

B. Output-feedback control
Consider a (dynamic) output feedback controller described by the difference equation of the form [5] with C(ξ) ∈ R m×m [ξ] and D(ξ) ∈ R m×p [ξ] polynomial matrices given by C(ξ) = I + C 1 ξ + C 2 ξ 2 + · · · + C n ξ l and D(ξ) = D 1 ξ + D 2 ξ 2 + · · · + D n ξ l . We define a state ζ c for (10) as ζ c := col(u(t−1), . . . , u(t−l), y(t−1), . . . , y(t−l)), yielding a state-space representation for the controller: Hence, the closed-loop system described by (3) and (11) has a representation With K := DC , the closed-loop system matrix A cl satisfies , we say that the controller (10) stabilizes (1) if the closed-loop system (12) is stable, i.e., if all eigenvalues of A z +B z K are in the open unit disk, since this implies stability of the closedloop system (1) and (10). The notion of stabilization with respect to the state-space representation (3) was introduced in [5] for data-driven stabilization. We note that in the singleinput-single-output case, (A z , B z ) is controllable if and only if A(ξ) and B(ξ) are coprime [5].

III. INFORMATIVITY FOR STABILIZATION
A. Informativity of input-output data Definition 1 The data (U, Y ) are said to be informative for quadratic stabilization by output-feedback controller (10) if there exist a K and P ≻ 0 so that (7)-(8), we find that the existence of K and P ≻ 0 so that is equivalent to the existence of K and P ≻ 0 such that (13) holds true. Now, for the data (U, Y ) to be informative for quadratic stabilization, we require the existence of K and P ≻ 0 so that (13) . This is precisely a problem that can be solved by the S-procedure; more specifically, by the matrix-valued S-lemma [4].
We remark that if there is a Z so thatZ ⊤ ΛZ ≻ 0 with Z := col(I, Z), called the generalized Slater condition [4], then (14) is also a necessary condition for informativity of input-output data for quadratic stabilization, if R − [Z ⊤ − U ⊤ − ] has full column rank (Lemma 1). Unlike in the case of inputstate data, which will be discussed next, we note that the generalized Slater condition can in general not hold true in the input-output case if l ≥ 1, since the noise affects a subspace of the extended state space, yielding a degenerate matrix Λ. The combination of noisy and noiseless states in ζ suggests that necessity could potentially be proven in general by a 'fusion' of the matrix S-lemma and matrix Finsler's lemma [12].

B. Informativity of input-state data
We will now consider a special case, where input-state data is available instead of input output data. That is, we measure a state y(t) = x(t) and the class of systems considered is with the corresponding data equation All systems that explain the data (U − , X) for some E − satisfying the cross-covariance bound (6) are in the set (6) and (16) hold}.
By (16), the set of feasible systems is Σ RQ  (6), we observe that for this special choice R − = I, we recover the set of feasible systems in [4], and, hence, the informativity conditions in [4].

Definition 2
The data (U − , X) are said to be informative for quadratic stabilization by state feedback if there exist a feedback gain K and P ≻ 0 so that We will now provide a necessary and sufficient condition for informativity of input-state data for quadratic stabilization, given prior knowledge on the cross-covariance (6). Consider the generalized Slater condition Proposition 1 Suppose that there exists a Z so that (18) holds true. Then the data (U − , X) are informative for quadratic stabilization if and only if there exist L ∈ R m×n , P ≻ 0, α ≥ 0 and β > 0 so that  Proof: (⇐) This is proven by the same argument as in the proof of Theorem 1. (⇒) Let the data be informative for quadratic stabilization, i.e., there exist K and P ≻ 0 so that, with Π defined in (13) with J 1 = 0, J 2 = 0: We will now show that ker Λ X 22 ⊆ Λ X 12 , such that necessity follows by the matrix S-lemma [4]. First, notice that ker Λ was chosen arbitrary, this shows that ker Λ X 22 ⊆ Λ X 12 . By ker Λ X 22 ⊆ Λ X 12 and (18), there exist α ≥ 0 and β > 0 so that, by [4,Theorem 13]: which is equivalent to (19) for L := KP by the Schur complement. This completes the proof.

IV. INCLUDING PERFORMANCE SPECIFICATIONS
We will now consider the problem of finding a controller (10) for which the closed-loop system achieves an H ∞ or H 2 performance bound from the input-output data (U, Y ). Consider the performance output z, given by z(t) = C z ζ(t) + D z u(t). For any pair (A z , B z ), the controller (10) yields the closed loop system Hence, the transfer matrix from e to z is given by for which the H ∞ and H 2 norm are denoted T H∞ and T H2 , respectively.
For given K, the H ∞ norm of T is less than γ, T H∞ < γ, if and only if there exists X ≻ 0 such that [13, p. 125 where A K := A z + B z K and C K := C z + D z K.

Definition 3
The data (U, Y ) are said to be informative for common H ∞ control by output-feedback controller (10) with performance γ if there exist a K and X ≻ 0 so that Theorem 2 The data (U, Y ) are informative for common H ∞ control with performance γ if there exist L ∈ R m×n , P ≻ 0, α ≥ 0 and β > 0 so that (21) holds true.
Proof: By a congruence transformation of (20) with diag(P, I, P, I) with P := X −1 and the application of the Schur complement (twice), the existence of K and X ≻ 0 so that (20) holds, is equivalent to the existence of P and L so that P ≻ 0 and Hence, the data (U, Y ) are informative for common H ∞ control with performance γ if and only if there exist P ≻ 0 and L such that P − γ −1 F ⊤ F ≻ 0 and (23) holds for all (Λ e , B e ) ∈ Σ RQ (U,Y ) . By assumption, there exist P ≻ 0, L, α ≥ 0 and β > 0 such that (21) holds true. By the Schur complement, (21) is equivalent to Π H∞ − αΛ βI 0 0 0 , which implies that (23) holds for all (Λ e , B e ) ∈ Σ RQ (U,Y ) . The conditions (21) are linear with respect to P , L, α and β. By a straightforward additional application of the Schur complement, (21) can also be made linear with respect to γ.
For a given controller parameter matrix K, the H 2 norm of T is less than γ, T H2 < γ, if and only if there exists X ≻ 0 such that [14, Proposition II.1], cf. [4] trace H ⊤ z XH z < γ 2 and X ≻ A ⊤ K XA K + C ⊤ K C K . (24) Definition 4 The data (U, Y ) are said to be informative for common H 2 control by output-feedback controller (10) with performance γ if there exist a K and X ≻ 0 so thatΣ RQ (U,Y ) ⊆ {(A z , B z ) | (24) holds true}.
Theorem 3 The data (U, Y ) are informative for common H 2 control with performance γ if there exist L ∈ R m×n , symmetric Z, P ≻ 0, α ≥ 0 and β > 0 so that trace Z < γ 2 , (25) holds true, Proof: By a congruence transformation of (24) with P := X −1 and the Schur complement (in both directions), it follows that (24) is equivalent to P − F ⊤ F ≻ 0, and trace H ⊤ z P −1 H z < γ 2 . Now, we can rewrite (27) as  which, by (7)- (8), holds if and and only if  There exist P ≻ 0, L so that (28), P − F ⊤ F ≻ 0 and trace H ⊤ z P −1 H z < γ 2 if and only if there exist P ≻ 0, L, Z so that (28), P − F ⊤ F ≻ 0, Z − H ⊤ z P −1 H z 0 and trace Z < γ 2 . Indeed, for Z := H ⊤ z P −1 H z we infer trace Z < γ 2 . Sufficiency follows from H ⊤ z P −1 H z Z ⇒ trace H ⊤ z P −1 H z ≤ trace Z. Hence, the data (U, Y ) are informative for common H 2 control with performance γ if and only if there exist P ≻ 0, L, Z so that Z −H ⊤ z P −1 H z 0, trace Z < γ 2 , P − F ⊤ F ≻ 0 and (28) hold for all (Λ e , B e ) ∈ Σ RQ (U,Y ) . By assumption, trace Z < γ 2 is satisfied, and consider a performance output z(t) = [0 0 1]x(t). The objective is to determine if input state data collected from the system are informative for common H 2 control. We consider a noise signal e(t) with a uniform distribution, taking values from the closed ball {e ∈ R 3 | e 2 2 ≤ 0.35}. First, we consider this noise bound to be known, represented by the noise model [4,Section VI.A]. This can be represented by the noise model (5) with R − = I, cf. [4,Equation (5)]. We consider the informativity analysis for various data lengths N ranging from N = 2 to N = 250. For each data length N , we generate 50 data sets. Given the bound on E − , we can verify informativity for common H 2 control via Theorem 17 in [4]. We find that the generalized Slater condition [4,Equation (16)], holds true for all data sets, thus the data are informative for common H 2 control with performance γ if and only if the condition [4, Equation (H 2 )] is feasible. The relative number of data sets for which this necessary and sufficient condition is feasible for some γ > 0 is visualized in Figure 1a for each data length N , in red. Naturally, if the condition is not feasible for any γ > 0, the data are actually not informative for feedback stabilization, although the true system is stable. Now, we consider the quadratic cross-covariance bound (5) for the noise. We choose an instrumental variable that contains lagged versions of the input: r(t) := col(u(t), u(t − 1), u(t − 2), . . . , u(t − 8), u(t − 9)). We assume prior knowledge in the sense that where H u is taken as H u = I, independent of N . The cross-covariance bounds hold true for all generated data sets. We verify that there exists some Z so that (18) holds true for all data sets. Hence, by Remark 2, the data are informative for common H 2 control with performance γ if and only if the conditions in Theorem 3 are feasible. The relative number of data sets for which this necessary and sufficient condition is feasible for some γ > 0 is visualized in Figure 1a for each data length N , in blue. For N ≥ 20, all data sets are informative for common H 2 control. For these data sets, the smallest H 2 norm upper bounds γ 2 are visualized in Figure 1b, where the median performance is indicated by a solid line and the shaded area is bounded by the 25 th and 75 th percentiles. In comparison, the H 2 norm that can be achieved by a state feedback controller with knowledge of (A 0 , B 0 ) is equal to 1.000, which therefore is a benchmark that cannot be outperformed by any data-based controller. Now, consider that noisy output measurements are available instead of state measurements. Consider system (1) with A(q −1 ) and B(q −1 ) such that T 0 (q −1 ) = A −1 (q −1 )B(q −1 ) with T 0 := C 0 (qI −A 0 ) −1 B 0 , where C 0 is the output matrix. We consider three cases: For each choice of output, we generate 50 data sets for data lengths ranging from N = 2 to N = 250. We choose an instrumental signal containing lagged input signals as before, which is therefore independent on the choice of output. The upper-bound is chosen H u = 0.3, which holds for all data sets. By Theorem 3, feasibility of the conditions for informativity for H 2 control for some γ > 0 is verified for each data set. The results are depicted in Figure 2. We observe that the data sets are not informative for low data lengths, which can be expected. For increasing data length, informativity becomes dependent on the choice of output. For N = 30, for example, 90% of the data sets yielded feasible informativity conditions for the choice of C 0 = [1 0 0], compared to less than 50% of the data sets for the other two choices for C 0 .

VI. CONCLUDING REMARKS
We have considered the problem of informativity of inputoutput data for control, with prior knowledge of the noise in the form of quadratic sample cross-covariance bounds. Sufficient informativity conditions for stabilization, H ∞ and H 2 control via dynamic output feedback were derived, which are also necessary if the state is measured. We have provided a numerical case study where data-informativity can be concluded with cross-covariance bounds, while the data are concluded to be non-informative with magnitude bounds. Finally, we have illustrated how the choice of output affects the informativity of input-output data via a numerical example.