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Sample Complexity for Evaluating the Robust Linear Observers Performance under Coprime Factors Uncertainty

This paper addresses the end-to-end sample complexity bound for learning in closed loop the state estimator-based robust H2 controller for an unknown (possibly unstable) Linear Time Invariant (LTI) system, when given a fixed state-feedback gain. We build on the results from Ding et al. (1994) to bridge the gap between the parameterization of all state-estimators and the celebrated Youla parameterization. Refitting the expression of the relevant closed loop allows for the optimal linear observer problem given a fixed state feedback gain to be recast as a convex problem in the Youla parameter. The robust synthesis procedure is performed by considering bounded additive model uncertainty on the coprime factors of the plant, such that a min-max optimization problem is formulated for the robust H2 controller via an observer approach. The closed-loop identification scheme follows Zhang et al. (2021), where the nominal model of the true plant is identified by constructing a Hankel-like matrix from a single time-series of noisy, finite length input-output data by using the ordinary least squares algorithm from Sarkar et al. (2020). Finally, a H-infinity bound on the estimated model error is provided, as the robust synthesis procedure requires bounded additive uncertainty on the coprime factors of the model.


Introduction
State estimation is a fundamental problem in control theory and machine learning. The utilization of state observers has been proven to be significant in both detecting and identifying faults in dynamical systems as well as monitoring and regulating those systems since the work of Luenberger (1966). The existence of disturbances and uncertainties provides significant difficulties in real-world applications, as practically all observer designs are based on the mathematical model of the plant. For this purpose, a number of sophisticated observer designs have been put out as solutions to the high-performance, robust observer-based regulator design challenge, which has lately attracted significant interest. The classical LQ control problem for LTI systems, served as the starting point for the aforementioned research problems, where the goal is to identify the best output feedback law that minimizes the expected value of a quadratic cost. In the past few years, significant research has been put into using modern statistical and optimization tools from the machine learning framework to approach classical control problems, see for instance Dean et al. (2018), Boczar et al. (2018), Mania et al. (2019), Dean et al. (2020), Zheng et al. (2020), Wang et al. (2015), Lee and Lamperski (2020), Tsiamis et al. (2020).
An end-to-end sample-complexity bound of learning observer-based H 2 controller for an unknown (potentially unstable) LTI plant that stabilizes the true system with high probability is established in this paper by incorporating recent advances in finite-time system identification. The resulting sub-optimal gap is bounded as a function of the level of model uncertainty. The end-to-end sample complexity bound for learning the robust observer-based H 2 controller is O where T is the time horizon for learning. Paper Organization: The paper is organized as follows: the general setup and problem formulation is given in Section II. The robust observer synthesis with uncertainty on the coprime factors is included in Section III. A brief discussion on the sub-optimality guarantees with end-to-end sample complexity results are stated in Section IV. Conclusion and future directions are given in Section V. All the proofs are postponed to the Appendices, where literature review, mathematical preliminaries and closed loop system identification scheme also have been discussed briefly.

General Setup and Technical Preliminaries
The notation used in this paper is fairly common in control systems. Upper and lower case boldface letters (e.g. G) are used to denote transfer function matrices, while lower and upper case letters (e.g. z and A) are used to denote vectors and matrices. The enclosed results are valid for discrete-time linear systems, therefore z denotes the complex variable associated with the Ztransform for discrete-time systems. A LTI system is stable if all the poles of its TFM are situated inside the unit circle for discrete time systems. The TFM of a LTI system is called unimodular if it is square, stable and has a stable inverse. For the sake of brevity the z argument after a transfer function may be omitted. R(z) denotes the set of all real-rational transfer functions and R(z) p×m denotes the set of p × m matrices having all entries in R(z). The notation T ε is used to indicate the mapping from signal ε to signal after combining all the ways in which is a function of ε and solving any feedback loops that may exist. For example, T zw is the mapping from the disturbances w to the regulated measurements z.

The State Estimation Problem
For a discrete-time LTI (Linear and Time Invariant) systems driven by Gaussian process and sensor noise, the state-space model is given by: where x k ∈ R n is the state of the system, u k ∈ R m is the control input and y k ∈ R p is the measured output and w k ∈ R m , δ k ∈ R n are the control additive and state additive disturbances, while ν k ∈ R p is the measurement noise, all considered to be Gaussian with zero mean and covariance matrices σ 2 w I, σ 2 δ I and σ 2 ν I respectively. A state estimator (observer) for (1) is defined as a system that provides an estimate x k of the internal state x k , while having access solely to the control input u and measured output y, with the underlying requirement that the estimation error converges to zero in the steady-state, that is lim where Ψ u (z) and Ψ y (z) are two LTI filters (stable Transfer Function Matrices (TFMs)) for the design of which one needs to know the model (1) of the plant, see for example Ding et al. (1994). The celebrated Kalman Filter, represents the canonical formulation of performance specifications for a state estimator (2) as it minimizes the transfer from the exogenous signals in (1) (e.g. the measurement noise ν k ) to the estimation error x k − x k (by using for example norm based costs).

Output Feedback Stabilizing Controllers
A standard unity feedback configuration is depicted in Figure 1 , where G ∈ R(z) p×m is a multivariable LTI plant and K ∈ R(z) m×p is an LTI controller. Here w, ν and r are the input disturbance, sensor noise and reference signal respectively while u, z and y are the controls, regulated signals and measurements vectors, respectively. If all the closed-loop maps from the exogenous signals [r T w T ν T ] T to any point inside the feedback loop are stable, then K is said to be an (internally) stabilizing controller of G or equivalently that K stabilizes G.

The Youla-Kuçera Parameterization of All Stabilizing Controllers
Theorem 2.2 (Ding et al. (1994), Vidyasagar (1985) Given a stabilizable and detectable statespace realization (1) of the plant G, then a DCF as in Definition 2.1 above is given by: where F and L are stabilizing state-feedback and estimation gains that allocate all eigenvalues of A F and A L inside the unit disk.
Remark 1 Theorem 2.2 above states that the DCF (4) of the plant is essentially equivalent with establishing certain stabilizing state-feedback F and estimation gain L, such that u Theorem 2.3 (Youla-Kucera) (Vidyasagar, 1985, Ch.5) Let M, N, M, N, X, Y, X, Y be a doubly coprime factorization of G. Any controller K Q stabilizing the plant G, can be written as where X Q , X Q , Y Q and Y Q are defined as: It also holds that K Q from (5) stabilizes G, for any stable Q.

Parameterization of All State Estimators
The following results provides the parameterization of all state observers of a given LTI system.
Theorem 2.4 (Ding et al. (1994)) Given stabilizing state-feedback F and estimation gain L, or equivalently, given a DCF (4) of the LTI plant (1) (see also Remark 1), let us denote P(z) (B) Furthermore, any state estimator for (1) can be written as where for some stable S(z) ∈ R(z) n×p . Conversely, for any stable S(z) it holds that (7), with (Ψ u S , Ψ y S ) as in (8), is a state estimator for (1).

Remark 2
The intrinsic connections of Theorem 2.4 with output feedback stabilization are evident, just as the affine parameterization (8) of all state-estimators is akin to the Youla parameterization of Theorem 2.3, but it is important to note that Theorem 2.4 holds just the same if the plant (1) is in open loop or if the plant is in a feedback interconnection with a stabilizing controller K. However, these two very distinct scenarios must be handled with care. In this paper we are interested in "learning" the optimal state estimator of an unknown plant in closed feedback loop. To this end the following two results (for the closed-loop scenario) will be instrumental towards the main result and surprisingly enough, they cannot be found in the original work from Ding et al. (1994).
Theorem 2.5 Consider the the LTI plant (1) in feedback interconnection with the controller given by: u k = F x k in tandem with any state-estimator of the form x = Ψ u S (z)u(z) + Ψ y S (z)y(z). The closed loop maps from the disturbances w and measurement noise ν to the estimation error (x − x) are affine functions of the S parameter from Theorem 2.4 (B), moreover: Theorem 2.6 Given a DCF (4) of the LTI plant (1) and its subsequent stabilizing state-feedback gain u k = F x k , let us assume that the F ∈ R m×n matrix is left invertible (it has full column rank). Then any stabilizing output feedback controller K Q from (5) can be realized as: Remark 3 The two theorems above clarify the fact that the two filters that realize any state estimator (7) in closed-loop are actually the closed loop maps from the exogenous signals to the estimation error. Furthermore, and this is important, under the assumption that the state-feedback gain matrix F is left invertible, there exists a bi-univocal relationship (10) between the S parameter from Theorem 2.4 and the Youla parameter. This immediately allows to rephrase parameterization (8) of all state observers which is affine in S, to a parameterization (Ψ u Q ; Ψ y Q ) affine in the Youla parameter, thus bridging the gap between any stabilizing controller K Q from (5) and its realization via: a fixed state-feedback gain F in tandem with the dynamic state estimator (Ψ u Q ; Ψ y Q ). The fixed state-feedback gain F comes from the initial stabilizing controller in the closed loop, as X(z) = −F (zI − A F ) −1 L is neither a function of Q nor a function of S.

A First Glimpse into the Separation Principle
We illustrate below the fact that any stabilizing controller can be realized either as a fixed statefeedback gain in tandem with a dynamic state-estimator or as a fixed estimation gain in tandem with dynamic state feedback. Both parameterizations are affine in the Youla parameter.

The Optimal State Estimator
THE PROBLEM: In this paper we consider the unknown plant (1) in feedback interconnection with some known stabilizing controller K, controller that is realized as: a fixed state-feedback gain F considered to be immutable, namely u k = F x k , in tandem with some state-estimator x = Ψ u u + Ψ y y. First we must learn the unknown system with high probability, in finite time, from a single trajectory in the closed loop. Finally, we must design the optimal state-observer that in tandem with the state-feedback gain u k = F x k yields the optimal LQG performance. Ding et al. (1994) and Subsection 2.4 Alazard and Apkarian (1999) Any stabilizing K Q from (5) can be realized via the static state-feedback gain F in tandem with the dynamic state estimator (Ψ u Q ; Ψ y Q ).
Any stabilizing K Q from (5) can be realized via the static estimation gain L in tandem with the dynamic feedback Q.
The canonical formulation of performance specifications for a state estimator is to minimize the transfer from the exogenous signals in to the estimation error. However, as stated above, the declared scope is to design a state-estimator specifically tailored to work in tandem with the fixed state-feedback gain u k = F x k . In this context, the choice of the optimality criterion is essential, as outlined below.
Proposition 2.7 We define the Optimal Observer Evaluation Problem, given a fixed state-feedback gain F with u = F x as: min which turns out to be equivalent with: Proof of Theorem 2.4, Theorem 2.5, Theorem 2.6 and Proposition 2.7 is provided in Appendix B.
Remark 4 (Estimation Error) The reason behind choosing (11) for the observer design in this context is mainly caused by the fact that the model of the plant can never be determined with absolute accuracy, since any learning algorithm produces outcomes which are inherently uncertain. Furthermore, the objective function from(11) pertains to the difference in H 2 performance in the closed loop between the state-feedback control u = F x (with direct access to the state) and any output feedback controller K Q . The thorough reasoning for this and all the underlying implications are deferred to Appendix C.

Robust Controller Synthesis: An Observer Based Approach
The outcome of the "learning" of the true plant G from closed-loop measurements comes in the form of a left coprime factorization of what we have dubbed the nominal model 1 , namely For the detailed description of the learning algorithm we refer to the Appendix G from Zhang et al. (2021). In order to evaluate the discrepancy between the learned G md and the true plant, we make a recourse to the preeminent method for modelling uncertainty for LTI systems (stemming from classical robust control), specifically via additive perturbations on the coprime factors.
Remark 5 On top of being able to cope with learning unstable plants (in closed loop), this method of modelling uncertainty, explicitly avoids the need of knowing apriori the McMillan degree (i.e. the state dimension of a minimal state-space realization) of the unknown plant, which is never known in practice. The flip side of this coin, is that since the learned nominal model G md and the true plant will not even have the same McMillan degree, it is impossible to retrieve anything about the state representation (1) of the true plant solely from the knowledge of G md . Consequently designing an optimal, robust state-estimator for the true plant only on the basis of learned nominal model appears to be a daunting task.
With the DCF of the nominal model of the plant write the Bézout's identity that incorporates the coprime factorization of the initial, known stabilizing controller Definition 3.1 (Model Uncertainty Set) The γ-radius model uncertainty set for the nominal plant G md with ∆ M , ∆ N both stable is defined as: (14) Definition 3.2 (γ-Robustly Stabilizing) A fixed stabilizing controller K of the nominal plant is said to be γ-robustly stabilizing iff K stabilizes not only G md but also all plants G ∈ G γ .
Assumption 1 It is assumed that the true plant, denoted by G pt , belongs to the model uncertainty In the presence of additive uncertainty on the coprime factors the Bézout's identity in (13) no longer holds, however, the following holds for certain stable ∆ M , ∆ N factors: The block diagonal structure of the right hand side term in (15) from (15) is unimodular i.e. it is square, stable and has a stable inverse.
2. The controller with which the closed-loop learning is being performed is assumed to be known.
Theorem 3.4 The Youla parameterization yields a γ-robustly stabilizing controller K Q iff its cor- The proofs for Lemma 3.3 and Theorem 3.4 are given on Appendix B. As an intermediary result, by employing Theorem 3.4 and the standard inequality from Appendix A it is concluded that: Starting from the left coprime factorization of the true plant, known to be of the form G pt = factors available from the known controller. Here, Φ 11 , Φ 22 are as in (15). By re-establishing the Bézout identity we are able to formulate the robust version of (12) as: Theorem 3.5 The Robust Linear Observer Evaluation Problem given a fixed state feedback gain F reads : Remark 6 (Validation of Constraints) We need the initial controller in the closed loop to be robust enough to maintain the mapping between S and Q as in Theorem 2.6. Necessarily, it is considered It can be seen that (17) is actually phrased in terms of the coprime factors of the true plant, which can never be learned in practice. The standard min-max formulation from Theorem 3.5 for the robust observer evaluation is non-convex by the fact that Φ −1 11 is no longer an affine function of Q. The duality gap renders the attempt to solve Q by switching the order of min and max impossible. In order to circumvent this, an upper bound on the cost functional will be derived and we formulate the robust observer evaluation problem in a Quasi-convex manner.
Proposition 3.6 (Quasi-Convex Formulation) For the true plant, G pt ∈ G γ the robust observer evaluation problem in (17) admits the following upper bound: The inner objective function in (18) is affine in Q, hence the inner optimization problem in Proposition 3.6 is convex for each fixed α. Proof for Proposition 3.6 is provided in Appendix B.
Remark 7 The Quasi-convex problem in Proposition 3.6 is formulated in frequency domain. To solve it in practice, we need to perform a Finite-Impulse Response (FIR) truncation on Markov parameters of these systems. After the FIR truncation, for each fixed α ∈ [0, 1/γ), An equivalent Semi Definite Programming (SDP) can be formulated for the inner optimization problem, which would give us the vectorization of Markov parameters of the optimal Q * to Proposition 3.6. Details on the SDP formulation is provided in Appendix B.

Analysis of End-to-End Performance
The performance of the Robust Linear Observer from (17), together with the fixed state feedback u = F x will be considered in this section. Denote the H 2 -cost of applying the control inputs u = F x and u = F x by J u and J u respectively. Then it is shown in Appendix C that: where e k represents the k th standard basis vector in R m and δ t is the discrete Dirac impulse function.
Then, by the upperbound in Proposition 3.6, we get that: Where Q * is the optimizer to Proposition 3.6.
Specifically, if the fixed state feedback gain F happens to be the stabilizing Riccati state-feedback F opt , then by the virtue of separation principle the cost J u in (20) becomes the optimal H 2 -cost. In this case, (20) immediately gives a bound for the difference in H 2 -cost between the Robust Linear Controller designed from (17) and the optimal Linear Quadratic Regulator (LQR) for the true plant. This argumentation is deferred to Appendix C. Furthermore, from (20), it is evident that (J u − J u ) ∼ O( γ 1 − γα ) which indicates that the sample complexity relies heavily on α. In practice, it is impossible to examine uncountably many α's in [0, 1/γ), one should pick the value of α empirically each time when formulating an SDP.
Remark 8 (Feasibility) As α < 1 γ is picked manually each time to formulate a new SDP and the performance of the observer degrades much faster with a larger α, one would like α to be as small as possible. However, since α serves as the constraint in the Quasi-convex problem, a relatively small α may render the feasible set empty. This implies that the robust observer performance rely on the quality of the initial controller. A better initial controller would provide not only a better fixed feedback gain, but also a larger feasible set for the inner optimization in Proposition 3.6.
We integrate the above results with the system identification guarantees of Zhang et al. (2022), to provide end-to-end sample complexity bounds for learning the linear observers given a fixed feedback gain. Then following the system identification procedure with probability at least (1 − δ) where δ is the failure probability, it holds that Combining with the prerequisite for robustness analysis, −∆ M −∆ N ∞ < γ as in Assumption 1, it is reasonable to consider that the robustness radius γ is at the level O logT T . m + dp + log(T /δ) T .
Combining Theorem 4.1 with (20), it follows that with high probability the difference J u and J u behaves as

Conclusion and Future work
In this paper, we have provided the sample complexity bounds for an observer-based robust LQG regulator synthesis procedure for an unknown plant, where uncertainty is modeled as additive perturbations on the coprime factors. We combined finite-time, non-parametric LTI system identification (Sarkar and Rakhlin (2019)) with the Youla parameterization for observer performance evaluation given a fixed state feedback gain. As an opened avenue for future research is the online learning of the observer-based LQG controller under the same type of model uncertainty. One possible direction is to work out the sample complexity for online learning for: (a) the optimal state feedback (LQR) in tandem with (b) the optimal state-observer (Kalman Filter (Tsiamis et al. (2020))) for a potentially unstable system. Bruce Lee and Andrew Lamperski. Non-asymptotic closed-loop system identification using autoregressive processes and hankel model reduction. Appendix This appendix is divided into following parts. Appendix A presents a brief review of related works on observer parameterization. Also, a handful of mathematical preliminaries on norm identities and inequalities (Zhou et al. (1996)) are provided here. An overview of the closed loop mapping is stated in Appendix B along with robust synthesis proofs for Lemma 3.3, Theorem 3.4 and Theorem 3.5. This appendix section also completes the suboptimality guarantee proof in Proposition 3.6. Appendix C presents a brief overview on H 2 optimal cost. Non-aymptotic closed loop system identification (Sarkar et al. (2020)) is discussed in Appendix D.

Appendix A. Related Works
Recent years have seen a significant amount of research work focused on the finite time (nonasymptotic) learning of the optimal LQ regulator for a "unknown" plant utilizing the modern optimization methods and statistical tools from the learning framework. For related work on the Identification of Dynamical Systems, Controller Design, Robust Control and Optimal Control we refer to the Appendix A of Zhang et al. (2021).
Observer Design: Using the factorization technique for the parameterization of linear observers and associated estimation error dynamics is a classical result. This outcome offers a dual representation of the popular linear controller parameterization and offers fresh information on observer design (Ding et al. (1990)) that may be applied to both robust observer design and observer construction. Ding et al. (1994) outline and address issues with design and parameterization of robust linear observers in the frequency domain. Methods for determining any compensator's observer-based or LQG form with arbitrary order are explored in Alazard and Apkarian (1999). The performance and attributes of advanced state observers are compared in Wang and Gao (2003). These observers were first put out as a solution to the traditional observers' reliance on a precise mathematical representation of the plant, such as the Kalman filter and the Luenberger observer. Classical approaches to robust Kalman Filtering can be found in Xie and Soh (1994), Einicke and White (1999), Ghaoui andCalafiore (2001), Sayed (2001), Levy and Nikoukhah (2012), where parametric uncertainty is explicitly taken into account during the kalman filter synthesis procedure. The process of designing a Kalman filter for an unknown or partially observed autonomous linear time-invariant system has been discussed in Tsiamis et al. (2020), which was the first end-to-end sample complexity bounds for an unidentified system's Kalman filtering. The strategy of Luenberger observer is investigated to suggest a solution to the state and parameter estimation for dynamical systems in Alessandri and Coletta (2001)

B.1. Closed Loop Details
Proof for Theorem 2.4: Without loss of generality, by ommiting the additive disturbance δ k in the state equation (1) we get: Lets define a pseudo-state vector (z) by M(z) (z) = u(z). Then system (1) can be expressed as: Denote P(z) = (zI − A F ) −1 B. Then by applying the rules of connecting stable systems and By plugging back (22) into (21) and (2), we get the following: From (23), it's clear that x(z) = Ψ u (z)u(z) + Ψ y (z)y(z) is an observer of (1) if and only if : (i) the transfer function from (z) to x(z) matches with the transfer function from (z) to x(z) and (ii) the transfer functions from noises w(z) and ν(z) to the estimation error (x(z) − x(z)) are stable. Clearly, (i) gives us Ψ u (z)M(z) + Ψ y (z)N(z) = P(z). (ii) is always satisfied as Ψ u (z) and Ψ y (z) follow the parameterization in (8). Combining (i) and (ii), after inverse z-transform, it's evident that: which is the definition of an observer and this completes the proof.
Proof for Theorem 2.5: The transfer function matrices Ψ u S (z) and Ψ y S (z) in (8) are stable as S(z) ∈ R(z) n×p is stable. It's possible to use (8) instead of (6) since the parameterization in (8) would always satisfy (6) and reveal a valid observer for each stable S(z) ∈ R(z) n×p if the Bézout identity(3) holds. Now, we consider the transfer function from the noises to the estimation error. From (23), we can conclude that Proof for Theorem 2.6: Since Q ∈ R(z) m×p and S ∈ R(z) n×p are stable TFMs, without loss of generality it can be assumed that, for any onto linear mapping K ∈ R m×n , there exists E(z) ∈ R(z) m×p , such that To go from S(z) to Q(z), we need to solve E(z) for a pre-specified K ∈ R m×n . In order to get a meaniful result, here we choose that K = F . Then from (5) and (7), we get the following: The second inequality in (26) provides us that: It follows that −(I − F (PY + S N))X Q = F (PX − S M)Y Q . Solving this yields that for K = F in (25), the corresponding E(z) is just − X(z).Thus if F ∈ R m×n is onto, it's always possible to write FS(z) = Q(z) + X(z) which completes the proof.
Proof for Proposition 2.7: The Optimal Observer Evaluation Problem due to Remark 3 below is: Now, the transformed error for the observer evaluation problem above follow from the relationships FS(z) = Q(z) + X(z) = Q(z) + X Q (z) − M(z)Q(z) and FP(z) = M(z) − I m as below: Hence, the Optimal Observer Evaluation Problem have the form:

B.2. Robust Synthesis Details
The Bezout identity is retrieved before finding the closed loop maps associated with it by using (15) as below: Proof for Lemma 3.3: From DCF matrix Φ in (15), Next using Bezout identity for nominal model in (13) it follows that Proof of Theorem 3.4 directly follows from Zhang et al. (2021). Before giving the proof for Theorem 3.4, the small gain theorem is stated here.
Proof for Theorem 3.4: For any stable Q satisfying is unimodular (square and stable with a stable inverse) due to the fact that: is guaranteed to be stable via the Small Gain Theorem.
Conversely, if a Youla parameter Q yields a γ-robustly stabilizable controller of the nominal model then necessarily Then by the Spectral Mapping Theorem (Douglas, 1972, page 41-42) there must exist is not unimodular and consequently the Youla parameter Q does not produce an γ-robustly stabilizable controller, which is a contradiction. The proof ends.
Proof for Theorem 3.5: Let's denote Ψ u pt and −Ψ y pt as the transfer from noises to the estimation error when connecting the controller K We use the norm of transformed error F Ψ u pt −F Ψ y pt H 2 to reveal the observer performance in terms of the Youla parameter Q as below: The norm of these two transfers above are the objective function that we seek to minimize. The guarantees the robustness by providing the invertibility of Φ 11 , it would be the constraint in the robust observer evaluation problem.
To proof Theorem 3.6 an important lemma is stated below which is a standard result optimization.
Lemma B.2 (Dean et al. (2020)) For functions f : X → R and g : X → R and constraint set C ⊆ X, consider Assuming that f (x) ≥ 0 and 0 ≤ g(x) < 1, (∀)x ∈ C, this optimization problem can be reformulated as an outer single-variable problem and an inner-constrained optimization problem (the objective value of an optimization over the empty set is defined to be infinity): The equivalence here is established by considering dividing the feasible set C into two parts: {x ∈ C|g(x) ≤ δ} and {x ∈ C|δ < g(x) < 1}.
Proof for Proposition 3.6: The objective function in Theorem 3.5 admits the following upper bound: The last part of the upper bound above is formulated as follows: Proposition B.3 (SDP formulation for Proposition 3.6) For simplicity, the following is denoted: Note that all four time-domain representations above are assumed to be known. Y md (z), X md (z), Y md (z), X md (z), M md (z), N md (z), M md (z), N md (z) are the DCFs of the initial controller and the nominal model. The system I m can be viewed as a system whose Markov parameters are given by δ[j]I m , where δ[j] is the Dirac Impulse function with respect to j.
Next, the decision variable Q(z) is considered as Q(z) = n j=0 Q j z −j , define: where ⊗ denotes the Kronecker product, and by convention, it is considered that Q −1 = Q −2 = · · · = 0.
The Markov parameters of the system Y md Q (z) X md Q (z) can be written in the sense of convolution as: The first (n + 1) Markov parameters are taken after the convolution and denoted as:

Appendix C. Optimal Cost
If the pair (A, B) is controlable and the pair (A, C) is observable, then it is appropriate to introduce the time domain-representation of the H 2 cost as: y T t y t , w t = e k δ t , ν k = 0} Here in (31), e k represents the k th standard basis vector in R m and δ t = 1, t = 0 0, otherwise .