An Efficient Data-Driven Framework for Linear Quadratic Output Feedback Control

TL;DR

This paper introduces a scalable, data-driven framework for designing optimal output feedback controllers for linear systems with unknown parameters, improving efficiency and applicability over existing methods.

Contribution

It develops a novel theoretical framework for state parameterization that handles unknown states and system matrices, enabling model-free policy iteration algorithms with guarantees.

Findings

Proposes a new state parameterization framework meeting Willems' Fundamental Lemma conditions.

Designs efficient model-free algorithms with theoretical guarantees for optimal output feedback control.

Demonstrates improved implementation, data efficiency, and convergence over existing methods.

Abstract

Linear quadratic regulator with unmeasurable states and unknown system matrix parameters better aligns with practical scenarios. However, for this problem, balancing the optimality of the resulting controller and the leniency of the algorithm's feasibility conditions remains a non-trivial challenge, as no well-established general method has yet been developed to address this trade-off. To address this gap, this study first develops a comprehensive theoretical framework for state parameterization that equivalently substitutes for unknown states. By analyzing the controllability of consistent systems satisfied by substitute states, this framework quantifies the capability of substitute state data matrices to parameterize unknown closed-loop systems and output feedback controllers, thereby constructing a modified state parameterization form that meets the complete data parameterization condition of Willems' Fundamental Lemma. Leveraging this framework, this study proposes efficient model-free off-policy policy iteration and value iteration algorithms with theoretical guarantees to solve for the optimal output feedback controller. Compared with existing studies, particularly for multi-output problems where existing model-free reinforcement learning algorithms may fail, the proposed method removes redundant information in substitute states and the additional full row rank condition on regression matrices, thereby ensuring the solution of optimal output feedback controllers equivalent to optimal state feedback controllers for multi-output systems. Furthermore, this study pioneers a comprehensive and highly scalable theoretical analysis of state parameterization from a data-driven viewpoint, and the proposed algorithms exhibit significant advantages in implementation conditions, data demand, unknown handling, and convergence speed.

Tables

Table 1. Table 1 : Efficiency comparison of PI Algorithms.

	LS-based PI	Algorithm 1
Average running time (s)	0.0095	0.0022
Average number of iterations	3.04	3.04

Table 2. Table 2 : Efficiency comparison of VI Algorithms.

	LS-based VI	Algorithm 2
Average running time (s)	0.2994	0.0068
Average number of iterations	73.08	36.68

Table 3. Table 3 : Robustness of Algorithm 1 to different W max W_{\max} .

$W_{\max }$	${10}^{-3}$	${10}^{-4}$	${10}^{-6}$
${\Delta }_K$	$2.5521\times {10}^{-7}$	$6.7282\times {10}^{-4}$	$6.8279\times {10}^{-4}$
${\Delta }_{xPx}$	$-0.1323$	$-6.1031\times {10}^{-4}$	$6.2246\times {10}^{-6}$

Table 4. Table 4 : Robustness of Algorithm 2 to different W max W_{\max} .

$W_{\max }$	${10}^{-3}$	${10}^{-4}$	${10}^{-6}$
${\Delta }_K$	$8.0905\times {10}^{-9}$	$8.835\times {10}^{-9}$	$4.3409\times {10}^{-8}$
${\Delta }_{xPx}$	$0.1498$	$0.0162$	$1.6355\times {10}^{-4}$

Equations

$${\mathcal{T}}_N(A,B,C):=\left[\begin{array}{cccc}0& & & \\ CB& 0& & \\ ⋮& \ddots & \ddots & \\ C{A}^{N-2}B& \cdots & CB& 0\end{array}\right];$$

$$x_{t+1}$$

$$J(x_0,u)$$

$$u^{\ast }=\arg \underset{u\in {\mathcal{U}}_{ad}}{\min }J(x_0,u).$$

$$P^{K_x}=(A+B{K}_x{)}^{\top }{P}^{K_x}(A+B{K}_x)+Q_x+K_x^{\top }R{K}_x.$$

$$K_x=-(B^{\top }{P}^{K_x}B+R{)}^{-1}{B}^{\top }{P}^{K_x}A.$$

$$P^{K_x^{\ast }}=Q_x+A^{\top }{P}^{K_x^{\ast }}A-A^{\top }{P}^{K_x^{\ast }}B(R+B^{\top }{P}^{K_x^{\ast }}B{)}^{-1}{B}^{\top }{P}^{K_x^{\ast }}A.$$

$$P^{K_y}=(A+B{K}_{y}C{)}^{\top }{P}^{K_y}(A+B{K}_{y}C)+Q_x+C^{\top }{K}_y^{\top }R{K}_{y}C.$$

$$P^{K_y^{\ast }}=-A^{\top }{P}^{K_y^{\ast }}B(R+B^{\top }{P}^{K_y^{\ast }}B{)}^{-1}{B}^{\top }{P}^{K_y^{\ast }}A+Q_x+A^{\top }{P}^{K_y^{\ast }}A+G^{\top }(R+B^{\top }{P}^{K_y^{\ast }}B)G.$$

$$P^{K_y^{\ast }}-P^{K_x^{\ast }}=$$

$$+(K_x^{\ast }-K_y^{\prime }{)}^{\top }(R+B^{\top }{P}^{K_x^{\ast }}B)(K_x^{\ast }-K_y^{\prime })+G^{\top }(R+B^{\top }{P}^{K_y^{\ast }}B)G,$$

$$\hat{K}:x_{t+1}^c=A_K{x}_t^c+B_K{y}_t,u_t=C_K{x}_t^c,$$

$${\hat{x}}_{t+1}=\left[\begin{array}{cc}A+B{C}_K& -B{C}_K\\ A-B_{K}C+B{C}_K-A_K& A_K-B{C}_K\end{array}\right]{\hat{x}}_t:={\mathcal{A}}_{\hat{K}}{\hat{x}}_t;$$

$$c(y_t,u_t)={\hat{x}}_t^{\top }\left[\begin{array}{cc}{Q}_x+C_K^{\top }R{C}_K& -C_K^{\top }R{C}_K\\ -C_K^{\top }R{C}_K& C_K^{\top }R{C}_K\end{array}\right]{\hat{x}}_t:={\hat{x}}_t^{\top }{\mathcal{Q}}_{\hat{K}}{\hat{x}}_t.$$

$$x_{t+1}^c=(A+B{K}_x-LC)x_t^c+L{y}_t,u_t=K_x{x}_t^c,$$

$$X_0$$

$$U_0$$

$$X_1$$

$$\left[\begin{array}{c}{\mathcal{H}}_N(u_{[0,T-1]})\\ {\mathcal{H}}_N(y_{[0,T-1]})\end{array}\right]\alpha =\left[\begin{array}{c}{\bar{u}}_{[0,N-1]}\\ {\bar{y}}_{[0,N-1]}\end{array}\right].$$

$$\mathrm{im}([I_n,K_x^{\top }{]}^{\top })\subset \mathrm{im}([X_0^{\top },U_0^{\top }{]}^{\top }).$$

$$x_t=[\mathcal{R}(A,B)-A^N{\mathcal{O}}^{†}(A,C)\mathcal{T}(A,B,C),A^N{\mathcal{O}}^{†}(A,C)]{\xi }_t:=M{\xi }_t.$$

$$x_t=A^N{x}_{t-N}+\mathcal{R}(A,B)u_{[t-N,t-1]},$$

$$y_{[t-N,t-1]}=\mathcal{O}(A,C)x_{t-N}+\mathcal{T}(A,B,C)u_{[t-N,t-1]}.$$

$$M\left[\begin{array}{cc}{I}_{mN}& 0\\ \mathcal{T}(A,B,C)& I_{pN}\end{array}\right]=[\mathcal{R}(A,B),A^N{\mathcal{O}}^{†}(A,C)],$$

$$\Omega:=\left[\begin{array}[]{c:c}\begin{array}[]{c:c}&I_{m(N-1)}\\ \hline\cr\mathbf{0}_{m\times m}&\end{array}&\mathbf{0}_{mN\times pN}\\ \hline\cr\mathbf{0}_{pN\times mN}&\begin{array}[]{c:c}&I_{p(N-1)}\\ \hline\cr\mathbf{0}_{p\times p}&\end{array}\end{array}\right],$$

$$x_t=(M+M_0){\xi }_t,$$

$${\xi }_{t+1}$$

$${\xi }_t=\left[\begin{array}{cc}{I}_{mN}& 0\\ \mathcal{T}(A,B,C)& \mathcal{O}(A,C)\end{array}\right]\left[\begin{array}{c}{u}_{[t-N,t-1]}\\ x_{t-N}\end{array}\right]:={\mathcal{M}}_N{\bar{x}}_{t-N},$$

$$\mathrm{im}({\mathcal{M}}_N)=\mathrm{im}(R({\mathcal{A}}_m,{\mathcal{B}}_m)).$$

$$\begin{split}&\operatorname{rank}\left(\begin{bmatrix}U_{0}\\ \Xi_{0}\end{bmatrix}\right)=\operatorname{rank}\left(\begin{bmatrix}\mathcal{H}_{1}(u_{[0,T-1]})\\ \mathcal{H}_{N}(u_{[-N,T-2]})\\ \mathcal{H}_{N}(y_{[-N,T-2]})\end{bmatrix}\right)=\operatorname{rank}\left(\begin{bmatrix}\mathcal{H}_{N+1}(u_{[-N,T-1]})\\ \mathcal{H}_{N}(y_{[-N,T-2]})\end{bmatrix}\right)\\ &=\operatorname{rank}\left(\underbrace{\left[\begin{array}[]{c:c}I_{m(N+1)}&\mathbf{0}\\ \hline\cr\begin{array}[]{c:c}\mathcal{T}(A,B,C)&\mathbf{0}\end{array}&\mathcal{O}(A,C)\end{array}\right]}_{\mathcal{M}_{N+1}}\begin{bmatrix}\mathcal{H}_{N+1}(u_{[-N,T-1]})\\ \mathcal{H}_{1}(x_{[-N,T-N-2]})\end{bmatrix}\right)=\operatorname{rank}(\mathcal{M}_{N+1})\\ &=m(N+1)+n\leq m(N+1)+pN.\end{split}$$

$$x_{t+1}^c=(A-LC)x_t^c+B{u}_t+L{y}_t,$$

$${ϵ}_{t+1}=(A-LC){ϵ}_t.$$

Full text

Keywords: direct data-driven control, reinforcement learning, linear-quadratic optimal control, output feedback, state parameterization.

1 Introduction

Linear quadratic regulator (LQR) is a foundational optimal control method that enables stable and optimal control of linear systems by optimizing a quadratic performance index under measurable states. It plays a critical role in ensuring precise and efficient process control across key fields, such as robotic control [1], aircraft control [2], and autonomous driving [3]. However, in practice, LQR faces two realities that are more aligned with real-world applications yet are inherently challenging: 1) the matrix parameters of linear systems are seldom known a priori; 2) system states often cannot be directly measured due to cost-effectiveness and engineering practicality constraints (for instance, only output data is transmitted in remote network transmission to protect sensitive information [4]). These realities underscore the significant practical value of research on model-free output feedback LQR. Moreover, there exist key theoretical hurdles that need to be addressed. First, since system states are unmeasurable, the state-feedback-related Riccati equation that characterizes the optimal LQR controller becomes unsolvable. In such cases, even if controllers are derived via static output feedback (SOF) or dynamic output feedback under the separation principle [5], they do not necessarily yield the optimal solution to LQR problem [6, 7, 8]. Furthermore, unknown system matrix parameters add a further layer of difficulty to the theoretical solution. In this combined scenario, independently designing state observers for dynamic output feedback controllers becomes extremely challenging. Notably, while some model-free algorithms adopt policies that redundantly utilize output data and can yield optimal solutions, their convergence conditions for multi-output (MO) systems remain overly strict, even to the point of being unachievable.

Some existing data-driven methods have attempted to address this LQR problem with unmeasurable states and unknown system parameters. However, first-principles modeling or system identification is prone to complex operations and high costs; furthermore, the lack of a separation principle between system identification and model-based controller design may result in suboptimal control policies [9]. Utilizing policy optimization algorithms in reinforcement learning (RL) to solve optimal SOF or dynamic controllers encounters disconnected feasible regions and complex optimization structures, making it difficult to guarantee algorithmic optimality [6, 7]. While calculating substitute vectors for unmeasurable states and directly applying them to RL-based policy iteration (PI) or value iteration (VI) algorithms can ensure the optimality, such approaches remain largely infeasible for MO problems, which stems from their neglect of the intrinsic structure among available data and reliance on least squares (LS) numerical methods [10, 8]. Within the behavioral systems framework [11], characterizing unknown systems and controllers via Willems’ Fundamental Lemma [12] enables more comprehensive integration of data information to overcome limitations of traditional methods; however, to date, such techniques still struggle to extend to MO scenarios [13].

To address these gaps, this paper proposes general and efficient direct data-driven algorithms that address the potential failure of the aforementioned RL algorithms in MO problems and extend the applicability of Willems’ Fundamental Lemma to MO settings, while ensuring controller optimality. More specifically, to obtain an efficient substitute form for unmeasurable states, we first develop a comprehensive and highly scalable state parameterization framework along with its corresponding control theory. Within this framework, we revisit two types of equivalence relationships between substitute state vectors (generated from input-output data) and unmeasurable states, and focus on analyzing the capacity of the substitute state data matrices to parameterize unknown closed-loop systems and output feedback controllers. This analysis yields a key conclusion: the data parameterization ability is strongly correlated with the controllability of the consistent systems exhibited by the substitute states. Building on this conclusion, we eliminate redundant information in substitute states and propose a modified state parameterization method that not only ensures complete data parameterization but is also guaranteed to apply to MO problems, thereby enabling the feasibility of Willems’ Fundamental Lemma in MO settings. Second, utilizing the substitute state data matrix generated via the proposed state parameterization, we re-represents the Bellman iteration equation and further proposes off-policy PI and VI algorithms for solving the output feedback LQR problem. These algorithms exhibit higher efficiency in terms of data demand, handling of unknowns, and convergence speed. Moreover, as we leverage the data matrix to fully exploit the representation capability of Willems’ Fundamental Lemma, there is no need to convert the iteration equations into LS problems for solution, which avoids additional data requirements for regression matrices—requirements that are difficult to satisfy in MO settings.

1.1 Related Works

(i) Behavioral Systems Theory. Behavioral systems theory [11] focuses on system trajectories and eliminates the need for a state-space representation, with each input-output trajectory termed a “behavior”. As the core result of this theory, Willems’ Fundamental Lemma demonstrates that for a controllable discrete-time linear time-invariant (DT-LTI) system, any valid trajectory can be expressed as a linear combination of time-shifted persistently exciting (PE) measured trajectories [12]. This lemma has found extensive applications in autonomous driving [14], power grid [9] and quadrotor control [15]. Leveraging subspace relationships, the research in [13] demonstrates that under a specific rank condition, closed-loop systems and controllers can be parameterized via input-state data. For single-output (SO) systems, sufficiently long historical input-output trajectories can directly replace unknown states without altering the rank condition [13], though this conclusion cannot be readily extended to MO systems. To address MO problems, the work in [16] proposes an intuitive data matrix construction method for characterizing unknown systems by leveraging Gauss transformations. However, this approach lacks comprehensive control-theoretic foundations and is restricted to DT systems.

(ii) Model-Free State Feedback LQR. Model-free state feedback results serve as foundational and inspirational references for model-free output feedback LQR problems. In contrast to linear matrix inequalities methods [17, 18], more “brute-force” RL approaches may offer better computational efficiency. The work in [19] reformulates LQR as an optimization problem and establishes the gradient domination condition, a key criterion for the global convergence of policy gradient. Building on this result, the research in [20] applies Willems’ Fundamental Lemma to propose the data-enhanced policy optimization algorithm for adaptively solving state feedback LQR problems. Studies in [21] and [22] develop model-free algorithms for stochastic state feedback LQR via PI and VI, respectively. Furthermore, the research in [23] integrates Willems’ Fundamental Lemma to propose an off-policy Q-learning algorithm for state feedback LQR, characterized by fewer unknowns and higher computational efficiency.

(iii) Output Feedback LQR. Regarding model-based results, the work in [24] provides the necessary and sufficient condition for solving the optimal SOF controller, and the work in [7] concludes that the cost of dynamic output feedback controllers varies with similarity transformations. Owing to the complex topological structure of output feedback LQR [25], gradient domination conditions are hard to satisfy for both SOF and dynamic output feedback, resulting in a lack of theoretical guarantees for the global convergence of policy gradient [7, 6]. An alternative approach is to reconstruct or estimate states, which can also be applied to model-free problems. For instance, the research in [26] uses historical input-output trajectories as a substitute for states in PI and VI; studies in [10, 27] employ data generated by a user-defined internal model to replace states in off-policy PI and VI, though the iteration process may be affected by observation errors. This issue is addressed in [8, 2] through incorporating observation error replication into the internal model. However, the aforementioned value-based iterative methods focus more on algorithm design and fail to address MO problems comprehensively; moreover, as these algorithms rely on LS, they may require more stringent data conditions and incur higher computational costs. This paper aims to address these issues; meanwhile, the results obtained may contribute to the implementation of policy optimization in output feedback LQR [28].

1.2 Comparisons and Contributions

This paper investigates the data-driven output feedback LQ optimal control. Compared with existing results, the main contributions of this paper are as follows.

• (i)

A general framework for learning optimal output feedback controllers is established using only input-output data. This framework exhibits high applicability and scalability, stemming from three key advantages: first, it eliminates additional data requirements; second, it is applicable to both SO and MO problems; third, it can be easily extended to other optimal control problems such as continuous-time settings.

This framework obviates the need for multiple attempts to ensure the full row rank property of the regression matrix in [8] and is guaranteed to solve MO problems optimally. It also addresses the challenge of extending the state feedback results in [23] and [22] to output feedback scenarios.

• (ii)

A generalized state parameterization method is proposed along with the corresponding control-theoretic support, demonstrating that the controllability of consistent systems is pivotal to the full characterization of output feedback controllers using input-output data. This state parameterization method enables seamless integration with Willems’ Fundamental Lemma and RL algorithms, making it applicable to both SO and MO problems. This theoretical support not only facilitates efficient data-based characterization of closed-loop systems and output feedback controllers in noise-free settings but also guides the enhancement of algorithmic robustness under small noise perturbations.

Notably, the findings in [16] and the vector autoregressive framework with exogenous input (VARX) [29] can be regarded as special cases of this work. In contrast, the proposed state parameterization framework is more general and supported by a comprehensive theoretical basis.

• (iii)

Data-efficient off-policy PI and VI algorithms are proposed for solving the optimal output feedback controller of LQR problems in a model-free setting, breaking through the traditional paradigm of dependence on LS. Moreover, the corresponding complete proof system for stability, convergence, optimality, and robustness is established.

Compared with [8], in terms of data utilization, the proposed algorithms relax the rank conditions on data matrices required for convergence in [8], and require a smaller amount of data; in terms of computational efficiency, the proposed algorithms involve fewer unknown parameters and exhibit faster convergence rates.

The rest of the paper is organized as follows. Section 2 introduces the output feedback LQR problem, along with model-based LQR results and Willems’ Fundamental Lemma. Section 3 presents the generalized state parameterization method and the corresponding control theory, and establishes results on data-parameterized output feedback controllers. Section 4 proposes efficient PI and VI algorithms that are applicable to MO problems. Section 5 gives the robustness analysis and more detailed discussions. Section 6 demonstrates the feasibility and advantages of the proposed algorithms through numerical experiments. Finally, the paper ends with the conclusion in Section 7.

Notations. $\mathbb{R}$ denotes the set of real numbers. $\mathbb{S}^{n}$ denotes the set of $n\times n$ symmetric matrices; $\mathbb{S}^{n}_{++}$ ($\mathbb{S}^{n}_{+}$) is defined as the set of $n\times n$ positive (semi-)definite matrices. $A\succ B(A\succeq B)$ means that matrix $A-B$ is positive (semi-)definite; $A^{\frac{1}{2}}$ denotes the unique symmetric positive semi-definite square root of $A\in\mathbb{S}_{+}^{n}$. $I_{n}$ is defined as a $n\times n$ identity matrix, and $\mathbf{0}$ denotes a zero vector or matrix with proper dimension. Let $\|\cdot\|$ denote the vector Euclidean norm or matrix spectral norm. For matrix $A$, $\operatorname{rank}(A)$, $\rho(A)$, $A^{\top}$, $A^{\dagger}$, $\sigma_{\min}(A)$, $\sigma_{\max}(A)$, $\mathrm{im}(A)$, and $\mathrm{(left)ker}(A)$ denote its rank, spectral radius, transpose, Moore-Penrose pseudo-inverse, minimum singular value, maximum singular value, image space, and (left-)kernel space, respectively. For square matrix $B$, $\mathrm{det}(B)$ and $\mathrm{adj}(B)$ denote its determinant and adjugate matrix, respectively; $\lambda(B)$ is defined as the vector consisting of all the eigenvalues of $B$. For real symmetric matrix $C$, $\lambda_{\min}(C)$ and $\lambda_{\max}(C)$ denote its minimum and maximum eigenvalues, respectively. Define $\mathrm{diag}(A,B)$ as the block diagonal matrix with main matrix blocks $A$ and $B$. $\mathcal{Z}(\cdot)$ denotes the z-transform; $\mathcal{Z}^{-1}(\cdot)$ denotes the inverse z-transform. For $A=(a_{1},\cdots,a_{m})\in\mathbb{R}^{n\times m}$ and $B=(b_{ij})\in\mathbb{S}^{n}$, define $\mathrm{vec}(A):=(a_{1}^{\top},\cdots,a_{m}^{\top})^{\top}\in\mathbb{R}^{mn}$ and $\mathrm{vech}(B):=[b_{11},\sqrt{2}b_{12},\cdots,\sqrt{2}b_{1n},b_{22},\sqrt{2}b_{23},\cdots,\sqrt{2}b_{n-1,n},b_{nn}]^{\top}$. The notation $\otimes$ is defined as the Kronecker product. For a signal sequence $\{u_{t}\}$, define its stacked window vector as $u_{[i,j]}:=[u_{i}^{\top},u_{i+1}^{\top},\cdots,u_{j}^{\top}]^{\top}$ with $i<j$. Accordingly, a Hankel matrix of depth $N$ is defined as $\mathcal{H}_{N}(u_{[0,T-1]}):=[u_{[0,N-1]},u_{[1,N]},\cdots,u_{[T-N,T-1]}]$ with positive integers $N,T$ and $N<T$. A standalone triple $(\mathcal{A},\mathcal{B},\mathcal{C})$ represents a DT-LTI system. Define $\mathcal{R}_{N}(A,B):=[A^{N-1}B,\cdots,AB,B]$, $\mathcal{O}_{N}(A,C):=[C^{\top},(CA)^{\top},\cdots,(CA^{N-1})^{\top}]^{\top}$, and

[TABLE]

the subscript $N$ will be omitted when the context is clear, and the notations will be simplified to $\mathcal{R}(A,B)$, $\mathcal{O}(A,C)$ and $\mathcal{T}(A,B,C)$.

2 Problem Formulation and Preliminaries

This section introduces the problem formulation, the results of model-based LQR, and the outcomes of direct data-driven control.

2.1 Problem Formulation

Consider a DT-LTI system of the form

[TABLE]

with unmeasurable state $x_{t}\in\mathbb{R}^{n}$, measurable input $u_{t}\in\mathbb{R}^{m}$ and output $y_{t}\in\mathbb{R}^{p}$. The constant matrices $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$ and $C\in\mathbb{R}^{p\times n}$ denote the dynamics matrix, input matrix and output matrix, respectively, with $p\leq n$. Without loss of generality, it is assumed that $C$ has full row rank, otherwise, the output contains redundant information. It is necessary to define an admissible controller set $\mathcal{U}_{ad}$ based on the adopted controller structure to guarantee the stability of the target controller.

Define an infinite horizon quadratic cost functional as

[TABLE]

with constant cost weighting matrices $Q\in\mathbb{S}^{p}_{+}$ and $R\in\mathbb{S}^{m}_{++}$. The function $c(y_{t},u_{t})$ denotes the one-step cost when using the input $u_{t}$ at state $x_{t}$. Additionally, $Q_{x}:=C^{\top}QC\in\mathbb{S}^{n}_{+}$ can be expressed as the state cost weighting matrix.

This paper aims to address the model-free output feedback LQR problem, which is described as follows.

Problem 2.1**.**

Considering system (1), where $A$, $B$, $C$ are unknown and states $\{x_{t}\}$ are unmeasurable, find an optimal control policy $u^{*}$ that satisfies

[TABLE]

We introduce the following standard assumption throughout this paper.

Assumption 2.2** (Controllability and observability).**

The pair $(A,B)$ is controllable, and the pairs $(A,C)$, $(A,Q^{\frac{1}{2}})$ are observable.

Remark 2.3**.**

In general, the stabilizability of $(A,B)$ suffices for the existence of optimal feedback controller [30]. Since this paper employs the data parameterization method based on Willems’ Fundamental Lemma, we assume $(A,B)$ is controllable here; this is consistent with key literature in this field [11, 13, 23]. Relaxation of this assumption will be further discussed in subsequent sections. The observability of $(A,C)$ guarantees that the input-output data of system (1) contains all the information about unknown states. From Lyapunov control theory, the observability of $(A,Q^{\frac{1}{2}})$ is a prerequisite for the convergence of iterative-based schemes [31].

2.2 Model-Based LQR

This subsection assumes that system matrices $A$, $B$, $C$ are known.

(i) State feedback controller. When the states are measurable, i.e., $C=I_{n}$, Problem 2.1 is equivalent to finding the optimal state feedback controller of the form $u_{t}=K_{x}x_{t}$, where $K_{x}$ is referred to as the state feedback gain. Substituting this feedback law into system (1), the closed-loop system is $x_{t+1}=(A+BK_{x})x_{t}$, the admissible control set is $\mathcal{U}_{ad}=\{K_{x}|\rho(A+BK_{x})<1\}$, and the cost functional (2) can be rewritten as the quadratic form $J(x_{0},K_{x})=x_{0}^{\top}P^{K_{x}}x_{0}$, where $P^{K_{x}}\in\mathbb{S}^{n}_{++}$ is the unique positive definite solution to the following Lyapunov equation:

[TABLE]

The corresponding greedy optimal gain is

[TABLE]

Then, the optimal state feedback controller is uniquely characterized by the discrete algebraic Riccati equation (DARE), i.e., $u^{*}=K_{x}^{*}x$, $K_{x}^{*}=-(B^{\top}P^{K_{x}^{*}}B+R)^{-1}B^{\top}P^{K_{x}^{*}}A$, where $P^{K_{x}^{*}}\in\mathbb{S}^{n}_{++}$ is the unique positive definite solution to the following DARE:

[TABLE]

Directly solving the above nonlinear algebraic equation remains non-trivial. Instead, we can compute the cost parameter matrix $P^{K_{x}}$ of the current gain via equation (4), then perform policy improvement via equation (5). Starting from an initial stabilizing gain, model-based PI is guaranteed to converge to the optimal $P^{K_{x}^{*}}$ and $K^{*}_{x}$. Alternatively, based on fixed-point iteration theory and contractivity, model-based VI can be conducted via equation (6) to approximate the optimal gain.

Furthermore, if $C\neq I_{n}$ and is invertible, the optimal output feedback controller can be solved analogously to the state feedback case. For an underdetermined $C$, the following two typical controllers are often employed.

(ii) SOF controller. For the SOF controller $u_{t}=K_{y}y_{t}$, it holds that $x_{t+1}=(A+BK_{y}C)x_{t}$, $\mathcal{U}_{ad}=\{K_{y}|\rho(A+BK_{y}C)<1\}$, and the cost functional (2) can be rewritten as $J(x_{0},K_{y})=x_{0}^{\top}P^{K_{y}}x_{0}$, where $P^{K_{y}}\in\mathbb{S}^{n}_{++}$ is the solution to the Lyapunov equation

[TABLE]

However, solving for the optimal SOF gain $K_{y}^{*}$ requires considering the non-full column rank issue of the output matrix $C$ [24]. Specifically, there exists a matrix $G\in\mathbb{R}^{m\times n}$ such that $K_{y}^{*}C+(B^{\top}P^{K_{y}^{*}}B+R)^{-1}B^{\top}P^{K_{y}^{*}}A=G$, where $P^{K_{y}^{*}}$ is the solution to the following equation:

[TABLE]

The matrix $G$ can be regarded as the pseudo-inverse error, and the above equation describing the optimal SOF controller may not have a solution. If the solution exists, the obtained $u=K_{y}^{*}y$ might still be a suboptimal solution to Problem 2.1, as can be observed from the defining equation $K_{y}^{\prime}:=-(B^{\top}P^{K_{y}^{*}}B+R)^{-1}B^{\top}P^{K_{y}^{*}}A$ and the following equation [8]:

[TABLE]

and given $R\succ 0$, $P^{K_{x}^{*}}\succ 0$, $P^{K_{y}^{*}}\succeq P^{K_{x}^{*}}$ holds, which implies $J^{*}(x_{0},K_{y}^{*})\geq J^{*}(x_{0},K_{x}^{*})$.

(iii) Dynamic output feedback controller. Consider the full-order dynamic output feedback controller

[TABLE]

where $x^{c}_{t}\in\mathbb{R}^{n}$ is the state of the controller, $A_{K}\in\mathbb{R}^{n\times n}$, $B_{K}\in\mathbb{R}^{n\times p}$ and $C_{K}\in\mathbb{R}^{m\times n}$ are the controller parameter matrices. Denote $\epsilon_{t}:=x_{t}-x^{c}_{t}$ and $\hat{x}_{t}:=[x_{t}^{\top},\epsilon_{t}^{\top}]^{\top}$, then the corresponding closed-loop system is

[TABLE]

the admissible control set is $\mathcal{U}_{ad}=\{A_{K},B_{K},C_{K}|\rho(\mathcal{A}_{\hat{K}})<1\}$, and the one step cost is

[TABLE]

Therefore, the cost functional can be rewritten as $J(x_{0},\hat{K})=\hat{x}_{0}^{\top}P^{\hat{K}}\hat{x}_{0}$, where $P^{\hat{K}}\in\mathbb{S}^{2n}_{+}$ is the solution to the Lyapunov equation $P^{\hat{K}}=\mathcal{A}_{\hat{K}}^{\top}P^{\hat{K}}\mathcal{A}_{\hat{K}}+\mathcal{Q}_{\hat{K}}$. By the separation principle [5], the optimal dynamic output feedback controller satisfies the following standard form:

[TABLE]

where $K_{x}\in\mathbb{R}^{m\times n}$ and $L\in\mathbb{R}^{n\times p}$ are the state feedback and Luenberger observer gains rendering $A+BK_{x}$ and $A-LC$ Schur stable, respectively. However, an improperly chosen $L$ may induce observation errors, making the resulting optimal dynamic feedback controller a suboptimal solution to Problem 2.1. Specifically, by partitioning $P^{\hat{K}}$ into four blocks according to the structure of $\hat{x}_{t}$, the Bellman equation gives that $P^{\hat{K}}_{11}$ corresponds to $P^{K_{x}}$, $P^{\hat{K}}_{12}=\mathbf{0}$, and $P^{\hat{K}}_{22}$ is strictly positive definite when $A-LC\neq\mathbf{0}$, satisfying $P^{\hat{K}}_{22}=(A-LC)^{\top}P^{\hat{K}}_{22}(A-LC)+K_{x}^{\top}(R+B^{\top}P^{\hat{K}}_{11}B)K_{x}$ [8]. Therefore, $J(x_{0},\hat{K}^{*})\geq J(x_{0},K_{x}^{*})$.

Based on the above analysis, Section 3 will adopt a modified output feedback form that is equivalent to state feedback. While this form is comparable to dynamic output feedback, it offers stronger optimality guarantees and simpler design than SOF or direct use of dynamic output feedback.

2.3 Willems’ Fundamental Lemma

This subsection presents the results of direct data-driven control under state feedback. When $C=I_{n}$, and matrices $A$ and $B$ are unknown, it is assumed that noise-free input-state data can be collected from the real physical system or its simulations. Consider the state, input, subsequent state sequences of length $T$,

[TABLE]

System (1) can be equivalently represented as $X_{1}=AX_{0}+BU_{0}$. In the system identification method, we first need to estimates the unknown system matrices by solving the LS problem: $[\hat{A},\hat{B}]=\arg\min_{A,B}\|X_{1}-[A,B][X_{0}^{\top},U_{0}^{\top}]^{\top}\|$. The minimum-norm solution $[\hat{A},\hat{B}]=X_{1}\left([X_{0}^{\top},U_{0}^{\top}]^{\top}\right)^{\dagger}$ exists if and only if $[X_{0}^{\top},U_{0}^{\top}]^{\top}$ has full row rank, which can be ensured by applying PE inputs. However, due to inherent limitations of system identification as described in Section 1, we prefer to directly design the target controller from data.

Definition 2.4** (Persistently exciting (PE)).**

A signal sequence $u_{[0,T-1]}$ is PE of order $N$, if the Hankel matrix $\mathcal{H}_{N}(u_{[0,T-1]})$ has full row rank, where $T\geq(m+1)N-1$.

The Hankel matrix constructed from PE input-state data enables non-parametric full characterization of system (1).

Lemma 2.5** (Willems’ Fundamental Lemma [12]).**

Assume the DT-LTI system (1) is controllable. If the input sequence $u_{[0,T-1]}$ of system (1) is a PE signal of order $n+N$, with the corresponding state sequence $x_{[0,T]}$ and output sequence $y_{[0,T-1]}$, then the following assertions hold.

• (a)

$\operatorname{rank}\left(\begin{bmatrix}\mathcal{H}_{N}(u_{[0,T-1]})\\ \mathcal{H}_{1}(x_{[0,T-N-1]})\end{bmatrix}\right)=mN+n$.

• (b)

Assume $N\geq\ell$ with $\ell$ being the observability index of system (1). The sequence $(\bar{u}_{[0,N-1]},\bar{y}_{[0,N-1]})$ is an input-output trajectory of length $N$ for system (1), if and only if there exists a vector $\alpha\in\mathbb{R}^{T-N-1}$ such that

[TABLE]

When $N=1$, Hankel matrices degenerate into the form (8); assertion (a) is often used to perform data parameterization [13]. From the full row rank property of $[X_{0}^{\top},U_{0}^{\top}]^{\top}$, it follows that

[TABLE]

Let $X_{0}G=I_{n}$, then the closed-loop systems and controllers can be parameterized as $X_{1}G$ and $U_{0}G$, respectively. Assertion (b), a corollary of (a), not only indicates that the input-output Hankel matrix $[\mathcal{H}_{N}(u_{[0,T-1]})^{\top},\mathcal{H}_{N}(y_{[0,T-1]})^{\top}]^{\top}$ contains all the information of system (1), but also that any trajectory of system (1) can be expressed as a linear combination of known trajectories with sufficient information content.

When the state is unmeasurable in Lemma 2.5, we try to replace the state with output in assertion (a). However, as shown in Section 3, the input-output data matrix after direct substitution does not necessarily have full row rank, introducing significant challenges for data parameterization. Notably, the input-output data matrix always satisfies assertion (b), containing all the information characterizing system (1). Section 3 will synthesize these two assertions to explore how to balance the utilization of output data in the closed-loop systems.

3 State Parameterization

In this section, it is assumed that the state of system (1) is unmeasurable, while its input-output data can be collected from the real physical system or its simulations. To find an equivalent form of the unmeasurable state, we revisit two commonly used state parameterization methods for DT systems in Subsection 3.1 and Subsection 3.2, respectively; unlike existing studies, these subsections focus on their data parameterization capabilities and the underlying control theory. Building on these results, in Subsection 3.3, we further construct a modified state parameterization form $x_{t}=Fv_{t}$ (where $v_{t}$ serves as a substitute state) that satisfies the following two requirements, and this form has stronger data parameterization capabilities and greater practicality.

• •

Under the state parameterization $x_{t}=Fv_{t}$, the output feedback controller $u_{t}=Kv_{t}$ is equivalent to the state feedback controller $u_{t}=K_{x}x_{t}$, and the substitute state $v_{t}$ can be obtained using only input-output data of system (1).

• •

The input data matrix $U_{0}$ and the substitute state data matrix $V_{0}:=[v_{0},\cdots,v_{T-1}]$ can be used to directly parameterize the unknown system (1) and the output feedback gain $K$.

The work in [32] provides an answer to the first controller equivalence requirement: if the parameterization matrix $F$ has full row rank, the control policy $u_{t}=Kv_{t}$ derived from the output feedback learning algorithm will achieve the same performance as the state feedback control policy $u_{t}=K_{x}x_{t}$, thereby guaranteeing the optimality of the resulting controller. Additionally, some sufficient conditions are also provided to ensure that $F$ has full row rank, albeit these conditions involve user-defined parameter design considerations. In contrast, the state parameterizations presented in this section differ in that they necessarily satisfy the full row rank condition for $F$ under Assumption 2.2, as formally established in Theorem 3.2, Theorem 3.5, and Lemma 3.9 below.

The motivation for the second requirement is as follows: for MO systems, if we directly use the substitute state matrix $V_{0}$ to replace $X_{0}$ in Subsection 2.3 without inspection or processing, $V_{0}$ does not necessarily have full row rank, invalidating assertion (a) in Lemma 2.5. In this case, $V_{0}$ cannot fully characterize system (1) and the output feedback gain $K$, potentially causing non-convergence, numerical instability, and other issues during iterations. Therefore, the main contribution of this section is to establish verification conditions and processing methods for ensuring that $V_{0}$ has full row rank, thereby enabling complete data parameterization for both SO and MO problems, as Theorem 3.11 demonstrates. However, as Lemmas 3.4, 3.8, and 3.9 show, the root cause of $V_{0}$ lacking full row rank lies in the uncontrollability of the consistent system with $v_{t}$ as its state. To facilitate the analysis, we first introduce the concept of system consistency.

Definition 3.1** (Systems Consistency).**

A DT-LTI system $v_{t+1}=\mathcal{A}v_{t}+\mathcal{B}u_{t},y_{t}=\mathcal{C}v_{t}$ is said to be consistent with system (1) if for any input $\{u_{t}\}$, there exists a state trajectory $\{v_{t}\}$ such that the corresponding input-output trajectory $\{u_{t},y_{t}\}$ generated by this system is identical to that generated by system (1). In this setup, it can be briefly denoted that $(\mathcal{A},\mathcal{B},\mathcal{C})$ is consistent with system (1).

In summary, we expect the system satisfied by the substitute state $v_{t}$ to be consistent with system (1), and the data matrix $V_{0}$ to have full row rank. The conceptual block diagram of Section 3 is shown in Figure 1.

3.1 Trajectory-Based State Parameterization

It is well known that the state $x_{t}$ of system (1) can be recovered from the finite-length historical input-output trajectory $\xi_{t,N}:=[u_{t-N}^{\top},\cdots,u_{t-1}^{\top},y_{t-N}^{\top},\cdots,y_{t-1}^{\top}]^{\top}$ [26]. While $\xi_{t,N}$ depends on the time window length $N$, we will omit the subscript $N$ and denote it as $\xi_{t}$ for notational brevity.

Theorem 3.2** (State parameterization based on delayed input-output).**

Consider system (1). If Assumption 2.2 holds and $N\geq\ell$ with $\ell$ being the observability index of system (1), then the following assertions hold.

• (a)

The state $x_{t}$ of system (1) can be expressed in terms of the measurable historical input-output trajectory $\xi_{t}$ as

[TABLE]

• (b)

The parameterization matrix $M\in\mathbb{R}^{n\times(m+p)N}$ has full row rank.

Proof.

For assertion (a), the following equations can be derived from the expression of system (1):

[TABLE]

[TABLE]

Since system (1) is observable and $N\geq\ell$, $\mathcal{O}(A,C)$ has full column rank with a unique left inverse. Rearranging the above two equations yields equation (10).

For assertion (b), noting

[TABLE]

the full row rank property of $M$ is directly obtained from the controllability in Assumption 2.2. ∎

Remark 3.3**.**

Suppose system (1) is uncontrollable but stabilizable. If the dynamic matrix $A$ in system (1) is nonsingular, $M$ still has full row rank according to equation (11). If $A$ has zero eigenvalues, $M$ may not have full row rank; this is consistent with [32].

The following Lemma 3.4 shows that the expression between $x_{t}$ and $\xi_{t}$ is not necessarily unique, i.e., $M_{0}$ in equation (12) may not be a zero matrix; more importantly, when $p>1$ (i.e., in MO problems), the consistent system (13) with $\xi_{t}$ as the state is uncontrollable, and this is the key factor that prevents the data matrix from fully parameterizing the closed-loop system and the output feedback gain.

Lemma 3.4**.**

Define $\mathcal{B}_{m}:=[\mathbf{0}_{m\times m(N-1)},I_{m},\mathbf{0}_{m\times pN}]^{\top}$, $\Gamma:=[\mathbf{0}_{p\times(mN+p(N-1))},I_{p}]^{\top}$,

[TABLE]

$M_{0}$* as a matrix satisfying $\mathrm{ker}(M_{0})=\mathrm{im}(\mathcal{R}(\Omega+\Gamma CM,\mathcal{B}_{m}))$, and $\mathcal{A}_{m}:=\Omega+\Gamma C(M+M_{0})$. If Assumption 2.2 holds and $N\geq\ell$, the following assertions hold.*

• (a)

The relationship between the state $x_{t}$ of system (1) and the historical input-output trajectory $\xi_{t}$ can be expressed by the following equation:

[TABLE]

where $M$ is given by equation (10), and $M_{0}$ may not be a zero matrix.

• (b)

System

[TABLE]

is consistent with system (1), i.e., system (13) satisfies Definition 3.1.

• (c)

The controllability matrix $\mathcal{R}(\mathcal{A}_{m},\mathcal{B}_{m})$ of system (13) has rank $mN+n$. System (13) is controllable if and only if $p=1$, or $p>1$ and $pN=n$; otherwise, it is uncontrollable.

Proof.

From Theorem 3.2, it is easy to show that $(\Omega+\Gamma CM,\mathcal{B}_{m},CM)$ is consistent with system (1), and $\xi_{t}\in\mathrm{im}(\mathcal{R}(\Omega+\Gamma CM,\mathcal{B}_{m}))$. Then, $M_{0}\xi_{t}=\mathbf{0}$ is obtained by the definition of $M_{0}$, which proves assertion (a) and assertion (b).

For assertion (c), define the behavioral subspace spanned by input-output trajectories of system (1) with length $N$ as $\mathcal{V}_{N}$, and the state space of system (13) as $\mathcal{V}_{N}^{M}$.For any $\xi_{t}\in\mathcal{V}_{N}$, it holds that

[TABLE]

where $\mathcal{M}_{N}$ has full column rank, i.e., $\operatorname{rank}(\mathcal{M}_{N})=mN+n$. From the controllability in Assumption 2.2, $\bar{x}_{t-N}$ can take any value, then $\mathcal{V}_{N}=\mathrm{im}(\mathcal{M}_{N})$. Assume that the initial state of system (13) belongs to its controllable subspace, thus $\mathcal{V}_{N}^{M}=\mathrm{im}(\mathcal{R}(\mathcal{A}_{m},\mathcal{B}_{m}))$. Moreover, by virtue of the systems consistency in assertion (b), it is obvious that $\mathcal{V}_{N}=\mathcal{V}_{N}^{M}$, i.e.,

[TABLE]

Thus, $\operatorname{rank}(\mathcal{R}(\mathcal{A}_{m},\mathcal{B}_{m}))=\operatorname{rank}(\mathcal{M}_{N})=mN+n$. We know that a system is controllable if and only if the rank of its controllability matrix equals the dimension of the state. The state dimension of system (13) is $(m+p)N\geq mN+n$, with equality if and only if $pN=n$. ∎

Let’s turn attention to the substitute state matrix $\Xi_{0}:=[\xi_{0},\cdots,\xi_{T-1}]$. Firstly, from the behavioral space perspective of system (1), with the controllability assumption and PE inputs of order $N+1+n$, analogously to equation (14), it follows that

[TABLE]

Thus, when $p>1$, $\Xi_{0}$ does not necessarily have full row rank even though it contains all the information about system (1). Secondly, from the state space perspective of system (13) and by Lemma 3.4, the redundancy in the substitute state $\xi_{t}$ may render the consistent system (13) uncontrollable under the MO setting. This violates the condition of Lemma 2.5, and thus the full row rank property of $[\Xi_{0}^{\top},U_{0}^{\top}]^{\top}$ cannot be directly established. Generally speaking, direct data-driven methods may fail if $\Xi_{0}$ is directly used to replace $X_{0}$ to parameterize the controllers and the closed-loop systems.

3.2 Observer-Based State Parameterization

Since the derivation of state parameterization (10) is essentially based on the DT system equation (1), state parameterization (10) is dead-beat and cannot be extended to continuous-time problems [32]. This subsection considers another state parameterization method with a wider scope of application, which estimates the true state via a Luenberger observer

[TABLE]

where $x^{c}_{t}\in\mathbb{R}^{n}$ is the estimated state, and $L$ is a user-defined Luenberger observer gain. Let the characteristic polynomial of $A-LC$ be denoted as $\Lambda(z):=\mathrm{det}(zI_{n}-(A-LC)):=z^{n}+a_{n-1}z^{n-1}+\cdots+a_{1}z+a_{0}$, and the adjoint matrix of $zI_{n}-(A-LC)$ be denoted as $\mathrm{adj}(zI_{n}-(A-LC)):=D_{n-1}z^{n-1}+\cdots+D_{1}z+D_{0}$ with $D_{i}\in\mathbb{R}^{n\times n},i=0,\cdots,n-1$. Define $\mathcal{D}:=[D_{0},\cdots,D_{n-1}]$ and let $a_{n}=1$. Using the relation $\mathrm{adj}(zI_{n}-(A-LC))(zI_{n}-(A-LC))=\mathrm{det}(zI_{n}-(A-LC))I_{n}$, it holds that $D_{i}=\sum_{j=1}^{n-i}a_{n-j+1}(A-LC)^{n-i-j}$, $i=0,\cdots,n-1$.

Define the observation error as $\epsilon_{t}:=x_{t}-x_{t}^{c}$, which satisfies

[TABLE]

Due to $\epsilon_{t}=(A-LC)^{t}\epsilon_{0}$, the observer-based state estimation has no error only when all the eigenvalues of $A-LC$ are zero. Otherwise, when all the eigenvalues of $A-LC$ are less than $1$ in magnitude but not all zero, the observation error asymptotically converges to $\mathbf{0}$ only as time approaches infinity. Therefore, in order to estimate the true state $x_{t}=x_{t}^{c}+\epsilon_{t}$, we need to parameterize $x^{c}_{t}$ and $\epsilon_{t}$ separately using measurable input-output data of system (1). To clarify the subsequent derivation, define $A_{\epsilon}$ as a user-defined dynamic matrix that shares the same eigenvalues as $A-LC$,

[TABLE]

$\mathcal{A}_{s}:=\mathrm{diag}(I_{m+p}\otimes A_{s},A_{\epsilon})$, and $\mathcal{B}_{s}:=\mathrm{diag}(I_{m+p}\otimes b_{s},\mathbf{0}_{n\times 1})$. Replace $A-LC$ used in defining $\mathcal{D}$ with $A_{\epsilon}$, then define $\mathcal{D}^{\epsilon}$ analogously. For $i=1,\cdots,m$ and $j=1,\cdots,p$, let $u_{t}^{i}$ and $y_{t}^{j}$ denote the $i$-th component of $u_{t}$ and the $j$-th component of $y_{t}$, respectively, with their z-transforms denoted as $U^{i}(z)$ and $Y^{j}(z)$. Let $B_{i}$ and $L_{j}$ denote the $i$-th column of $B$ and the $j$-th column of $L$, respectively; define $S_{u}^{i}:=\mathcal{D}(I_{n}\otimes B_{i})$, $S_{y}^{j}:=\mathcal{D}(I_{n}\otimes L_{j})$,

[TABLE]

where $\eta_{t}^{u^{i}}$ and $\eta_{t}^{y^{j}}$ satisfy systems $\eta_{t+1}^{u^{i}}=A_{s}\eta_{t}^{u^{i}}+b_{s}u^{i}_{t}$, $\eta_{0}^{u^{i}}=\mathbf{0}$ and $\eta_{t+1}^{y^{j}}=A_{s}\eta_{t}^{y^{j}}+b_{s}y^{j}_{t}$, $\eta_{0}^{y^{j}}=\mathbf{0}$, respectively.

Theorem 3.5** (State parameterization based on input-output filtering).**

Consider system (1). If Assumption 2.2 holds, then the following assertions hold.

• (a)

The state $x_{t}$ of system (1) can be expressed in terms of the input-output filter vector $\eta_{t}\in\mathbb{R}^{(m+p+1)n}$ as

[TABLE]

where $\eta_{t}^{u}:=[(\eta_{t}^{u^{1}})^{\top},\cdots,(\eta_{t}^{u^{m}})^{\top}]^{\top}$, $\eta_{t}^{y}:=[(\eta_{t}^{y^{1}})^{\top},\cdots,(\eta_{t}^{y^{p}})^{\top}]^{\top}$, $S_{u}:=[S_{u}^{1},\cdots,S_{u}^{m}]$, $S_{y}:=[S_{y}^{1},\cdots,S_{y}^{p}]$, $\eta_{t}^{\epsilon}$ satisfies the system

[TABLE]

$S_{\epsilon_{x}}:=\mathcal{D}(I_{n}\otimes\epsilon_{0})$, $S_{\epsilon_{\eta}}:=\mathcal{D}^{\epsilon}(I_{n}\otimes\eta_{0}^{\epsilon})$, and $S_{\epsilon}:=S_{\epsilon_{x}}S_{\epsilon_{\eta}}^{-1}$. The substitute state $\eta_{t}$ can be generated in a model-free manner via the following user-defined system:

[TABLE]

• (b)

The parameterization matrix $S$ has full row rank.

Proof.

The proof of assertion (a) can be found in [8]. For the sake of completeness, we present this result here and specify the parameters in more detail.

We now present the proof of assertion (b), which differs from other references in stating that the parameterization matrix $S$ must have full row rank under the controllability in Assumption 2.2. Note that in equation (19), $\eta_{t}^{u}$ and $\eta_{t}^{y}$ are obtained by operating on each component of $u_{t}$ and $y_{t}$, respectively; whereas here, the operations are performed on the vectors as a whole. Applying an elementary row transformation matrix $P^{\eta}$ to $\eta_{t}$, where the matrix swaps the $i$-th and $(i+n)$-th rows of $\eta_{t}$, and the $(mn+j)$-th and $(mn+j+n)$-th rows of $\eta_{t}$, for $i=1,\cdots,m$ and $j=1,\cdots,p$, and it follows that

[TABLE]

and $x_{t}=[S_{u},S_{y},S_{\epsilon}](P^{\eta})^{\top}\sigma_{t}=\mathcal{D}[I_{n}\otimes B,I_{n}\otimes L,(I_{n}\otimes\epsilon_{0})(\mathcal{D}^{\epsilon}(I_{n}\otimes\eta_{0}^{\epsilon}))^{-1}]\sigma_{t}$. Since $\mathcal{D}^{\epsilon}(I_{n}\otimes\eta_{0}^{\epsilon})$ is a user-defined invertible matrix and each block of $\mathcal{D}$ is a linear combination of $(A-LC)^{i}$ for $i=0,\cdots,n-1$, we only need to focus on the matrix $S^{\prime}:=[(A-LC)^{n-1},\cdots,I_{n}](I_{n}\otimes[B,L,\epsilon_{0}])$. By contradiction, suppose $S^{\prime}$ does not have full row rank. According to the Hautus criterion, there exists a nonzero vector $q$ such that $q^{\top}[(A-LC)-\lambda I_{n},B,L,\epsilon_{0}]=\mathbf{0}$, where $\lambda$ is an eigenvalue of $A-LC$. This implies $q^{\top}(A-LC)=\lambda q^{\top}$, $q^{\top}B=\mathbf{0}$ and $q^{\top}L=\mathbf{0}$, thus $q^{\top}A=\lambda q^{\top}$, $q^{\top}[A-\lambda I_{n},B]=\mathbf{0}$, which contradicts the controllability assumption. Therefore, $S^{\prime}$ has full row rank, and so does $S$. ∎

Remark 3.6**.**

Suppose system (1) is uncontrollable but stabilizable. The sufficient condition in [32] can ensure the full row rank property of $\mathcal{D}(I_{n}\otimes L)$ by selecting $A-LC$ with eigenvalues distinct from those of $A$ to guarantee the controllability of $(A-LC,L)$. Alternatively, if the characteristic polynomial of $A-LC$ is its minimal polynomial, the $(A-LC)$-invariant subspace equals $\mathbb{R}^{n}$, rendering $\mathcal{D}(I_{n}\otimes\epsilon_{0})$ of full row rank; a practical approach is to assign distinct eigenvalues to $A-LC$ [8].

Remark 3.7** (Inclusion relation between two state parameterizations [32]).**

If $N=n$ in Theorem 3.2, all the eigenvalues of $A-LC$ are set to zero and $A_{\epsilon}=\mathbf{0}$ in Theorem 3.5, then $\Lambda(z)=z^{n}$, $\xi_{t}=P^{\eta}\eta_{t}=\sigma_{t}$, $MP^{\eta}=[S_{u},S_{y}]$, and state parameterizations (10) and (17) are equivalent.

Lemma 3.8 characterizes the controllability of the consistent system with $\eta_{t}$ as the state, and it shows that this system is uncontrollable when $p>1$ (i.e., in MO problems).

Lemma 3.8**.**

If Assumption 2.2 holds and the substitute state $\eta_{t}$ is generated by equation (19), then the following assertions hold.

• (a)

The relationship between the state $x_{t}$ of system (1) and the input-output filter vector $\eta_{t}$ can be expressed by the following equation:

[TABLE]

where $\mathrm{ker}(S_{0})=\mathrm{im}(\mathcal{R}(\mathcal{A}_{s}+\mathcal{B}_{s}^{y}CS,\mathcal{B}_{s}^{u}))$, $\mathcal{B}_{s}^{u}$ consists of the first $nm$ columns of $\mathcal{B}_{s}$, and $\mathcal{B}_{s}^{y}$ consists of columns $nm+1$ to $n(m+p)$ of $\mathcal{B}_{s}$.

• (b)

System

[TABLE]

is consistent with system (1), i.e., system (20) satisfies Definition 3.1.

• (c)

The rank of the controllability matrix of system (20) is $mn+\mathbf{1}_{\{\lambda(A-LC)\neq\mathbf{0}\}}n+n$, where $\mathbf{1}_{\{\lambda(A-LC)\neq\mathbf{0}\}}$ is an indicator function that takes the value 0 if all eigenvalues of $A-LC$ are equal to zero, and takes the value 1 otherwise. System (20) is controllable if and only if $p=1$.

Proof.

The proof of assertions (a) and (b) is similar to that in Lemma 3.4, so we only prove assertion (c). If all the eigenvalues of $A-LC$ are equal to zero, the assertion follows directly from Lemma 3.4. Next, we prove that the rank of $\mathcal{R}(\mathcal{A}_{m}+\mathcal{B}_{m}^{y}C(S+S_{0}),\mathcal{B}_{m}^{u})$ is $mn+2n$ when $A-LC$ has nonzero eigenvalues. Denote the operator

[TABLE]

The first $mn$ rows of $P^{\eta}$ are defined as $P^{\eta}_{u}$, and rows $mn+1$ to $(m+p)n$ are defined as $P^{\eta}_{y}$. From the forms of equations (17) and (19), it can be observed that the subsystems corresponding to $\eta_{t}^{u}$, $\eta_{t}^{y}$ and $\eta_{t}^{\epsilon}$ in system (20) are mutually independent. Therefore, the proof idea is as follows: first, discuss the state spaces of $\eta_{t}^{u}$, $\eta_{t}^{y}$ and $\eta_{t}^{\epsilon}$ respectively, and then discuss the sum of these three spaces.

The state space of $P_{u}^{\eta}\eta_{t}^{u}$ is isomorphic to that of $\eta_{t}^{u}$, and $P_{u}^{\eta}\eta_{t}^{u}=\mathcal{Z}^{tra}(u_{t})=[u_{t-n}^{\top},\cdots,u_{t-1}^{\top}]^{\top}-a_{n-1}P_{u}^{\eta}\eta_{t-1}^{u}-\cdots-a_{0}P_{u}^{\eta}\eta_{t-n}^{u}$. Due to $\eta_{0}^{u}=\mathbf{0}$, the state space of $P_{u}^{\eta}\eta_{t}^{u}$ coincides with the trajectory subspace of $[u_{t-n}^{\top},\cdots,u_{t-1}^{\top}]^{\top}$. A similar result holds for the state space of $\eta_{t}^{y}$. Denoting $\mathcal{G}:=\mathcal{T}(A,LC,C)\mathcal{O}(A-LC,I_{n})$, it follows that

[TABLE]

Therefore, the state space of $[(\eta_{t}^{u})^{\top},(\eta_{t}^{y})^{\top}]^{\top}$ coincides with $\mathrm{im}(\mathcal{M}_{N}^{\epsilon})$, and $mn+n\leq\operatorname{rank}(\mathcal{M}_{N}^{\epsilon})\leq mn+2n$; this is because the first $mn+n$ columns of $\mathcal{M}_{N}^{\epsilon}$ are linearly independent, and the linear dependency of the last $n$ columns is related to the state space of $\epsilon_{t}$.

Now, we consider the state space of $\eta_{t}^{\epsilon}$. Note that $\mathcal{D}^{\epsilon}(I_{n}\otimes\eta_{0}^{\epsilon})$ is a user-defined invertible matrix and a combination of $\mathcal{G}_{\epsilon}:=[A_{\epsilon}^{n-1}\eta_{0}^{\epsilon},\cdots,A_{\epsilon}\eta_{0}^{\epsilon},\eta_{0}^{\epsilon}]$, thus $\mathcal{G}_{\epsilon}$ is invertible. For the autonomous system (18), the state space of $\eta_{t}^{\epsilon}$ coincides with $\mathrm{im}(\mathcal{G}_{\epsilon})=\mathbb{R}^{n}$, which contains the state space of $\epsilon_{t}$.

Considering the sum of the state subspaces of $[(\eta_{t}^{u})^{\top},(\eta_{t}^{y})^{\top}]^{\top}$ and $\eta_{t}^{\epsilon}$, which equals $\mathrm{im}(\mathcal{R}(\mathcal{A}_{s}+\mathcal{B}_{s}^{y}C(S+S_{0}),\mathcal{B}_{s}^{u}))$, its dimension is $mn+2n$ according to the above analysis. The state dimension of the consistent system (20) is $mn+pn+n\geq mn+2n$, with equality if and only if $p=1$. ∎

Consider the substitute state matrix $H_{0}:=[\eta_{0},\cdots,\eta_{T-1}]$. Firstly, system (19) consists of a controllable canonical form system and an autonomous system, and its input also includes the output of system (1) and a zero vector. Even if the input of system (1) satisfies the PE condition, the output of system (1) will lose part of excitability after passing through the $n$-th order system. Consequently, system (19) is not fully excited, the state space of $\eta_{t}$ is a subset of $\mathbb{R}^{(m+p+1)n}$. Secondly, the consistent system (20) fails to meet the controllability under the MO settings, violating the condition of Lemma 2.5. Consistent with the conclusion in Subsection 3.1, $H_{0}$ in MO problems lacks full row rank and thus cannot directly replace the unknown $X_{0}$.

3.3 Projection-Based Data Efficient State Parameterization

Based on Subsection 3.1 and Subsection 3.2, neither state parameterization (10) nor (17) satisfies the complete data parameterization requirement initially proposed in Section 3, due to the uncontrollability of the consistent systems in MO problems. However, directly designing a state parameterization that satisfies the controller equivalence and controllability of consistent system is non-trivial, and we note that the controllable parts of consistent systems already contain all the information of system (1). Thus, this subsection modifies the state parameterizations (10) and (17) by projecting consistent systems onto their controllable subspaces to obtain full row rank substitute state matrices. For concise representation, set $N=n$ as in Subsection 3.1, and denote the state parameterizations (10) and (17) uniformly as $x_{t}=E\zeta_{t}$. Denote $n_{\zeta}$ as the dimension of $\zeta_{t}$, which takes the values $(m+p)n$ for (10) and $(m+p+1)n$ for (17). The corresponding consistent systems (13) and (20) are uniformly denoted as $(\mathcal{A},\mathcal{B},\mathcal{C})$, and the controllability matrix of $(\mathcal{A},\mathcal{B},\mathcal{C})$ has a rank of $n_{v}$, where $n_{v}$ equals $mn+n$ for (10) and $mn+\mathbf{1}_{\{\lambda(A-LC)\neq\mathbf{0}\}}n+n$ for (17).

Lemma 3.9** (Projection).**

Suppose the projection matrix $P\in\mathbb{R}^{n_{\zeta}\times n_{v}}$ satisfies

[TABLE]

If Assumption 2.2 holds and $\zeta_{t}=Pv_{t}$, then the state $x_{t}$ can be uniquely expressed in terms of $v_{t}$, i.e., $x_{t}=EPv_{t}:=Fv_{t}$, where the parameterization matrix $F$ has full row rank. Furthermore, the following system (22) with $v_{t}$ as the state

[TABLE]

is consistent with system (1) and is controllable, where none of $\mathcal{A}$, $\mathcal{C}$ and $E$ involves $M_{0}$ or $S_{0}$.

Proof.

It follows immediately from equation (21) that $M_{0}P=\mathbf{0}$ or $S_{0}P=\mathbf{0}$, and $\operatorname{rank}(P)=n_{v}$. Thus, the relation $x_{t}=Fv_{t}$ and the consistent system (22) hold. Since $E$ has full row rank from Theorem 3.2 or Theorem 3.5, and $P$ is a matrix with full column rank and $n_{v}\geq n$, it holds that $F=EP$ also has full row rank. Furthermore, since $\operatorname{rank}(\mathcal{R}(P^{\dagger}\mathcal{A}P,P^{\dagger}\mathcal{B}))=\operatorname{rank}(P^{\dagger}\mathcal{R}(\mathcal{A},\mathcal{B}))=\operatorname{rank}(\mathcal{R}(\mathcal{A},\mathcal{B}))=n_{v}$ with the last equality derived from Lemma 3.4 or Lemma 3.8, system (22) must be controllable. ∎

When $A$, $B$ and $C$ are unknown, computing $\mathcal{R}(\mathcal{A},\mathcal{B})$ is infeasible. In this case, the projection of consistent system can be instantiated as the Gauss transformation on the data matrix $Z_{0}:=[\zeta_{0},\cdots,\zeta_{T-1}]$.

Corollary 3.10** (Gauss transformation instance).**

Consider the substitute state matrix $Z_{0}:=[Z_{u}^{\top},Z_{y}^{\top},Z_{\epsilon}^{\top}]^{\top}$ with $Z_{u}\in\mathbb{R}^{mn\times T}$, $Z_{y}\in\mathbb{R}^{pn\times T}$, $Z_{\epsilon}\in\mathbb{R}^{n\times T}$. Suppose there exists a Gauss transformation matrix $G=[G_{1}^{\top},G_{2}^{\top}]^{\top}\in\mathbb{R}^{pN\times pN}$ with $G_{1}\in\mathbb{R}^{n\times pN}$ and $G_{2}\in\mathbb{R}^{(pN-n)\times pN}$ such that $Z_{y}^{1}:=G_{1}Z_{y}$ forms the row-linearly independent part of $Z_{y}$ and $Z_{y}^{2}:=G_{2}Z_{y}$ constitutes the remaining row-linearly dependent part. Let $P^{G}:=\mathrm{diag}(I_{mn},G_{1}^{\dagger},\mathbf{1}_{\{\lambda(A-LC)\neq\mathbf{0}\}}I_{n})$, then $P^{G}$ is a projection matrix satisfying equation (21).

Proof.

Since the state space of $\zeta_{t}$ coincides with $\mathrm{im}{\mathcal{R}(\mathcal{A},\mathcal{B})}$ and $P^{G}$ is a matrix composed of basis vectors of the state space of $\zeta_{t}$, equation (21) holds. ∎

Specifically, if $\zeta_{t}$ is derived from the state parameterization (10) as $\zeta_{t}=\xi_{t}$ and perform projection on $\zeta_{t}$ with $P^{G}$, then $x_{t}=[\mathcal{R}(A,B)-A^{N}(\mathcal{O}^{1}(A,C))^{\dagger}\mathcal{T}^{1}(A,B,C),A^{N}(\mathcal{O}^{1}(A,C))^{\dagger}]v_{t}$, where $\mathcal{O}^{1}(A,C)\in\mathbb{R}^{n\times n}$ and $\mathcal{T}^{1}(A,B,C)\in\mathbb{R}^{n\times mn}$ are composed of linearly independent rows of $\mathcal{O}(A,C)$ and $\mathcal{T}(A,B,C)$, respectively [16].

Since the consistent system (22) in Lemma 3.9 is necessarily controllable, the state parameterization $x_{t}=Fv_{t}$ definitely satisfies the requirements initially proposed in Section 3, whether for SO or MO problems. Then, we can establish the full row rank property and the direct data parameterization capability of the substitute state matrix $V_{0}:=[v_{0},\cdots,v_{T-1}]$ below.

Theorem 3.11**.**

For both SO and MO problems, if the input sequence $\{u_{t}\}$ of system (1) satisfies PE of order $2n+1$, the matrix $[V_{0}^{\top},U_{0}^{\top}]^{\top}$ generated under Lemma 3.9 has full row rank. Consequently, $[V_{0}^{\top},U_{0}^{\top}]^{\top}$ can be employed to characterize system (1) and directly parameterize the output feedback controller $u_{t}=Kv_{t}$, i.e., $\mathrm{im}\left([P^{\top},K^{\top}]^{\top}\right)\subseteq\mathrm{im}\left([V_{0}^{\top},U_{0}^{\top}]^{\top}\right)$.

Proof.

The full row rank property follows directly from Lemma 3.9 and Lemma 2.5. Below, we prove that the full row rank property of $[V_{0}^{\top},U_{0}^{\top}]^{\top}$ is a necessary and sufficient condition for directly parameterizing the feedback control gain $K$.

Suppose we do not apply projection to the uncontrollable consistent system $(\mathcal{A},\mathcal{B},\mathcal{C})$. Applying $u_{t}=K\zeta_{t}$ to system (1) yields $\zeta_{t+1}=(\mathcal{A}+\mathcal{B}K)\zeta_{t}:=\mathcal{F}\zeta_{t}$. Define the set $\Sigma_{0}:=\{(\mathcal{A}_{0},\mathcal{B}_{0})|\mathcal{A}_{0}Z_{0}+\mathcal{B}_{0}U_{0}=\mathbf{0}\}$ and $\mathcal{F}_{0}:=\mathcal{A}_{0}+\mathcal{B}_{0}K$. Note that for any positive constant $\alpha$, $\rho(\mathcal{F}+\alpha\mathcal{F}_{0})<1$ must be satisfied, from which it follows that $\rho(\frac{1}{\alpha}\mathcal{F}+\mathcal{F}_{0})<\frac{1}{\alpha}$. Taking the limit as $\alpha\rightarrow\infty$, we conclude that $\rho(\mathcal{F}_{0})=\mathbf{0}$. Since left-multiplying $\mathcal{F}_{0}$ by any matrix preserves the property that the resulting matrix has a spectral radius of zero, it follows that $\rho(\mathcal{F}_{0}^{\top}\mathcal{F}_{0})=\mathbf{0}$. Consequently, $\mathcal{F}_{0}=[\mathcal{A}_{0},\mathcal{B}_{0}][I,K^{\top}]^{\top}=\mathbf{0}$. However, since $Z_{0}$ does not have full row rank, there exists a nonzero matrix $\mathcal{A}_{1}$ such that $\mathcal{A}_{1}Z_{0}=\mathbf{0}$ and $(\mathcal{A}_{1},\mathbf{0})\in\Sigma_{0}$, contradicting $[\mathcal{A}_{1},\mathbf{0}][I,K^{\top}]^{\top}=\mathcal{A}_{1}=\mathbf{0}$. Thus, $\mathrm{ker}\left([Z_{0}^{\top},U_{0}^{\top}]^{\top}\right)\not\subseteq\mathrm{ker}\left([I_{n_{\zeta}},K^{\top}]^{\top}\right)$, i.e., $\mathrm{im}\left([I_{n_{\zeta}},K^{\top}]^{\top}\right)\not\subseteq\mathrm{im}\left([Z_{0}^{\top},U_{0}^{\top}]^{\top}\right)$. In this case, $[Z_{0}^{\top},U_{0}^{\top}]^{\top}$ cannot be used to fully parameterize the output feedback gain $K$.

Now, we consider the controllable consistent system (22). Applying $u_{t}=Kv_{t}$ to system (1) yields $v_{t+1}=P^{\dagger}(\mathcal{A}+\mathcal{B}KP^{\dagger})Pv_{t}:=\mathcal{F}v_{t}$. Analogously, define $\Sigma_{0}:=\{(\mathcal{A}_{0},\mathcal{B}_{0})|\mathcal{A}_{0}V_{0}+\mathcal{B}_{0}U_{0}=\mathbf{0}\}$ and derive $\mathcal{F}_{0}:=P^{\dagger}(\mathcal{A}_{0}+\mathcal{B}_{0}KP^{\dagger})P=P^{\dagger}[\mathcal{A}_{0},\mathcal{B}_{0}][P^{\top},K^{\top}]^{\top}=\mathbf{0}$. It is easy to prove that $(P^{\dagger}\mathcal{A}_{0},P^{\dagger}\mathcal{B}_{0})\in\Sigma_{0}$. From the full row rank property of $[V_{0}^{\top},U_{0}^{\top}]^{\top}$, it thus follows that $\mathrm{im}\left([P^{\top},K^{\top}]^{\top}\right)\subseteq\mathrm{im}\left([V_{0}^{\top},U_{0}^{\top}]^{\top}\right)$. Therefore, the output feedback gain $K$ can be directly parameterized by $[V_{0}^{\top},U_{0}^{\top}]^{\top}$. ∎

Lemma 3.12 below gives the relationship between the eigenvalues of the controllable consistent system (22) and system (1).

Lemma 3.12** (Eigenvalues).**

Considering the controllable consistent system (22), the eigenvalues of its dynamic matrix $P^{\dagger}\mathcal{A}P$ consist of the eigenvalues of $A$ and the user-defined eigenvalues of $A-LC$, where the eigenvalues of $A-LC$ are repeated $n+\mathbf{1}_{\{\lambda(A-LC)\neq\mathbf{0}\}}$ times.

Proof.

Since state parameterization (10) and consistent system (13) are special cases of (17) and (19), we only need to consider the case where $E=S$ and $\mathcal{A}=\mathcal{A}_{s}+\mathcal{B}_{s}^{y}CS$. Suppose $\lambda_{i}$ is an eigenvalue of $A$ with eigenvector $\phi_{i}$, $i=1,\cdots,n$. Let $\{u_{t}\}\equiv\mathbf{0}$ and $x_{0}=\phi_{i}$, thus $y_{t}=\lambda_{i}^{t}C\phi_{i}$. From system (22), it follows that

[TABLE]

Thus, $\lambda_{i}$ is an eigenvalue of the matrix $P^{\dagger}(\mathcal{A}_{s}+\mathcal{B}_{s}^{y}CS)P$ with eigenvector $(CSP)^{\dagger}C\phi_{i}$.

Now, we prove that the remaining eigenvalues of $P^{\dagger}(\mathcal{A}_{s}+\mathcal{B}_{s}^{y}CS)P$ equal those of $A-LC$. Suppose $\mu_{j}$ is an eigenvalue of $P^{\dagger}(\mathcal{A}_{s}+\mathcal{B}_{s}^{y}CS)P$ but not $A$ with eigenvector $\beta_{j}$, $j=1,\cdots,nm+n$. Let $\{u_{t}\}\equiv\mathbf{0}$ and $v_{0}=\beta_{j}$, thus $v_{t}=\mu_{j}^{t}\beta_{j}$ and $x_{t}=\mu_{j}^{t}SP\beta_{j}$. If $SP\beta_{j}\neq\mathbf{0}$, $\mu_{j}$ must be an eigenvalue of $A$, contradicting the assumption. From $SP\beta_{j}=\mathbf{0}$ and equation (20), it holds that $\mu_{j}\beta_{j}=P^{\dagger}(\mathcal{A}_{s}+\mathcal{B}_{s}^{y}CS)P\beta_{j}=P^{\dagger}\mathcal{A}_{s}P\beta_{j}$. From the block structures of matrices $\mathcal{A}_{s}$ and $\mathcal{B}_{s}$, together with Lemma 3.9, it follows that $\mathcal{B}_{s}^{y}CS$ and $P$ only modify the output-corresponding subsystems. Thus, the eigenvalues of the first $n$-th input-corresponding subsystems and the last error-corresponding subsystem in $P^{\dagger}(\mathcal{A}_{s}+\mathcal{B}_{s}^{y}CS)P$ still equal the eigenvalues of $A-LC$. With at most $mn+n$ eigenvalues satisfying $\mu_{j}\neq\lambda_{i}$, the proof is completed. ∎

Regarding the relationship between the real system (1) and consistent systems, Lemma 3.4 and Lemma 3.8 transform system (1) into consistent systems, while Lemma 3 in [29] employs VARX results to convert the consistent system (13) into a similarity transformation of system (1). Next, we discuss the relationship between several controllers.

First, consider the output feedback controller $u_{t}=Kv_{t}$ and the dynamic output feedback controller (7). Since (7) is also a dynamic system, interchanging $\{u_{t}\}$, $\{y_{t}\}$ and substituting $(A,B,C)$ with $(A_{K},B_{K},C_{K})$ in the above analysis, this shows that $u_{t}=Kv_{t}$ corresponds to a consistent system of (7). Define the dynamic matrix of the closed-loop system formed by (1) and (7) as $\mathcal{A}_{u}$. The closed-loop system formed by (22) and $u_{t}=Kv_{t}$ is $v_{t+1}=P^{\dagger}(\mathcal{A}+\mathcal{B}KP^{\dagger})Pv_{t}$. Analogously to Lemma 3.12, the eigenvalues of $P^{\dagger}(\mathcal{A}+\mathcal{B}KP^{\dagger})P$ consist of the eigenvalues of $\mathcal{A}_{u}$ and the user-defined eigenvalues of $A-LC$, where the eigenvalues of $A-LC$ are repeated $n+\mathbf{1}_{\{\lambda(A-LC)\neq\mathbf{0}\}}$ times. Thus, designing a stabilizing output feedback controller $u_{t}=Kv_{t}$ corresponds to designing a stabilizing dynamic output feedback controller (7).

Next, consider the output feedback controller $u_{t}=Kv_{t}$ and the state feedback controller $u_{t}=K_{x}x_{t}$. As discussed in the beginning of Section 3, since the parameterization matrix $F$ in $x_{t}=Fv_{t}$ must have full row rank, the output feedback controller $u_{t}=Kv_{t}$ and the state feedback controller $u_{t}=K_{x}x_{t}$ are equivalent. Furthermore, if $K_{x}$ is given, $K$ can be expressed as $K=K_{x}F+\Delta$ with $\Delta\in\mathrm{leftker}(F^{\dagger})$; if $K$ is given, $K_{x}$ can be expressed as $K_{x}=KF^{\dagger}$. Let $J(x_{0},K)=v_{0}^{\top}Pv_{0}$, then the equation

[TABLE]

holds and is equivalent to the Lyapunov equation (4) under $P=F^{\top}P^{K_{x}}F$, $K_{x}=KF^{\dagger}$ and the full row rank property of $F$. Thus, the costs in (2) under these two controllers are also equal, and the stabilizing output controller $u_{t}=Kv_{t}$ ensures $\rho(A+BKF^{\dagger})=\rho(A+BK_{x})<1$.

Up to these points, Section 3 has established a complete theoretical framework for the generalization of state parameterization, with this framework focusing more on the data representation capability of the substitute state. Based on these results, Problem 2.1 reduces to Problem 3.13 below.

Problem 3.13**.**

Consider system (1) with unknown $A$, $B$, $C$ and unmeasurable $\{x_{t}\}$. Find an optimal output feedback gain $K^{*}$ such that $\{u^{*}_{t}=K^{*}v_{t}\}$ satisfies equation (3), where the manner of generating $\{v_{t}\}$ satisfies Lemma 3.9.

4 Efficient Off-Policy Value-Based Iteration Algorithms

This section considers system (1) with unknown matrices $A$, $B$, $C$ and unmeasurable states. Leveraging the full row rank data matrices $[V_{0}^{\top},U_{0}^{\top}]^{\top}$ and $V_{1}:=[v_{1},\cdots,v_{T}]$ from Theorem 3.11, we designs efficient off-policy value-based iterative algorithms for Problem 3.13. The algorithm design demonstrates how the conclusions in Section 3 connect Willems’ Fundamental Lemma with PI and VI in output feedback problems; the algorithm analysis focuses more on elaborating the transformation between output feedback and state feedback results. The last subsection details the advantages of the proposed algorithms.

Under the output feedback controller $u_{t}=Kv_{t}$, the cost functional (2) can serve as the corresponding V-function:

[TABLE]

and the optimization problem is $\min_{K}V^{K}(x_{k})$. Similar to the cost functional (2), $V^{K}$ can also be expressed as the quadratic form $v_{k}^{\top}Pv_{k}$ with $P=F^{\top}P^{K_{x}}F$, and substituting it into (23) gives

[TABLE]

Define Q-function $Q^{K}(x_{k},u_{k})$ under $u_{t}=Kv_{t}$ as the value function that uses input $u_{k}$ at time $k$ and applies feedback input $u_{t}=Kv_{t}$ from time $k+1$

[TABLE]

the final equation implies that the Q-function can also be expressed in quadratic form. Moreover, the following relationships hold

[TABLE]

Note that the V-function does not involve the choice of control policy; hence the optimization problem that actually needs to be solved is $\min_{u_{t}}Q^{K}(x_{t},u_{t})=\min_{u_{t}}\psi_{t}^{\top}\Theta\psi_{t}$, whose optimal solution is $u_{t}^{*}=K^{*}x_{t}$ with

[TABLE]

Equation (25) represents a greedy policy, which is necessarily an improved policy compared to $K$, i.e., $V^{K^{*}}\leq V^{K}$. Substituting equation (25) into $P=[I_{n_{v}},K^{\top}]\Theta[I_{n_{v}},K^{\top}]^{\top}$ yields the updated $P^{up}$ as

[TABLE]

If both $P$ and $\Theta$ are treated as integral unknown parameters, we can utilize V-function (23), Q-function (24), policy improvement equation (25), and value update equation (26) for model-free PI or VI.

4.1 Q-Value Based Policy Iteration

From equation (24), it holds that

[TABLE]

When the current control gain $K$ is known, $\Theta$ is the only unknown in equation (27), thus equation (27) can be used to evaluate $Q^{K}$. Noting that the value of equation (27) is a scalar, but to solve for the matrix $\Theta$, we must solve a system of equations. To address this, we can use the input data matrix $U_{0}$ satisfying PE of order $2n+1$, the corresponding output data matrix $Y_{0}$ and the substitute state matrix $V_{0}$ under Theorem 3.11 to form the following system of equations:

[TABLE]

Here, $\Psi_{0}:=[\psi_{0},\psi_{1},\cdots,\psi_{T-1}]:=[V_{0}^{\top},U_{0}^{\top}]^{\top}$ has full row rank and can directly parameterize system (1) and control gain $K$, with $T\geq n_{v}+m$. Thus, equations (28) and (25) can serve as the policy evaluation and policy improvement steps in Q-value based PI. The complete procedure is given in Algorithm 1. Since the data matrices used are constructed from arbitrary input-output data satisfying the PE condition, the algorithm is off-policy.

Remark 4.1** (Initial stabilizing control gain).**

Similarly to [33], with $X_{0}$ replaced by $V_{0}$, we can use the following methods to find a stabilizing control gain as the initial gain. (a) Dead-beat method. Let $g(v)$ be the basis matrix of $\mathrm{ker}(V_{0})$. Design a control gain $K^{0}_{d}$ for the system with $V_{1}V_{0}^{\dagger}$ and $V_{1}g(v)$ as the dynamics matrix and the input matrix by placing poles at the origin, then $K^{0}:=U_{0}(V_{0}^{\dagger}+g(v)K^{0}_{d})$ can serve as the initial control gain in Algorithm 1. (b) Linear matrix inequalities method. If $\Sigma$ satisfies $V_{0}\Sigma=(V_{0}\Sigma)^{\top}$ and ${\small\begin{bmatrix}V_{0}\Sigma&V_{1}\Sigma\\ (V_{1}\Sigma)^{\top}&V_{0}\Sigma\end{bmatrix}}\succ 0$, then $U_{0}\Sigma(V_{0}\Sigma)^{-1}$ can serve as the initial control gain in Algorithm 1.

Define $\bar{F}:=\mathrm{diag}(F,I_{m})$, and let $\Theta_{K_{x}}$ be defined by $\Theta=\bar{F}^{\top}\Theta_{K_{x}}\bar{F}$. From the results in Subsection 3.3, i.e., $X_{0}=FV_{0}$, $K_{x}=KF^{\dagger}$ and $X_{1}=AX_{0}+BU_{0}$, equation (28) can be transformed into

[TABLE]

By the full row rank property of $[X_{0}^{\top},U_{0}^{\top}]^{\top}$, the above equation is equivalent to the Lyapunov equation:

[TABLE]

where $K_{AB}:=[I_{n},K_{x}^{\top}]^{\top}[A,B]$ is stable if and only if $A+BK_{x}$ is also stable [23]. Thus, the iteration essence of Algorithm 1 is equivalent to state feedback PI in [23]; however, Algorithm 1 can be applied to problems where the states are unknown. If the state of system (1) are known, Algorithm 1 can degenerate into the latter. Furthermore, the stability, convergence, and optimality of Algorithm 1 can be proven analogously to the results about state feedback PI in [23]. Notably, the proof process here will be affected by matrix $F$, particularly the convergence rate of the algorithm; hence, the proof of the following lemmas will focus on demonstrating this influence.

Lemma 4.2**.**

Consider Algorithm 1. If the initial feedback gain $K^{0}$ is stable, then equation (29) has a unique solution, all the feedback gains $K^{i}$ generated by Algorithm 1 are stable, and the sequence $\{\Theta^{i}\}$ is monotonically non-increasing, eventually converging to $\Theta^{*}$ that corresponds to the optimal feedback gain $K^{*}$. When $K^{i+1}$ satisfies the small threshold termination condition of Algorithm 1, it is an approximate optimal solution to Problem 3.13.

Proof.

Unlike [23], here we focus on how the assertions obtained under output feedback can be transformed into those under state feedback, and vice versa. The key to this lies in the flexible application of the full row rank property of $F$, as well as the stability results of the controllers presented in Subsection 3.3. Define $K_{AB}^{i}$ by substituting $K_{x}$ with $K_{x}^{i}$ in $K_{AB}$. Let $\Theta_{K_{x}}^{i+1}$ be defined by $\Theta^{i+1}=\bar{F}^{\top}\Theta_{K_{x}}^{i+1}\bar{F}$.

Assume $K^{i}$ is stable. From the analysis in Subsection 3.3, $K_{x}^{i}=K^{i}F^{\dagger}$ is unique and stable, thus $\rho(K_{AB}^{i})<1$. Replacing $\Theta_{K_{x}}$ with $\Theta^{i+1}_{K_{x}}$ in equation (31) yields that $\Theta^{i+1}_{K_{x}}$ is unique and positive definite. Furthermore, due to the full row rank property of $\bar{F}$, the solution $\Theta^{i+1}$ to equation (29) is unique. Since $K^{i+1}$ is the optimal solution to equation (29), it follows that

[TABLE]

[TABLE]

Similar to the analysis of equation (31), the above inequality is equivalent to the Lyapunov inequality $\Theta^{i+1}_{K_{x}}\geq\mathrm{diag}(Q_{x},R)+(K_{AB}^{i+1})^{\top}\Theta^{i+1}_{K_{x}}K_{AB}^{i+1}$. From the positive definiteness of $\Theta^{i+1}_{K_{x}}$, the stability of $K_{x}^{i+1}$ can be derived. Then, the stability of $K^{i+1}$ follows from the stability results in Subsection 3.3. Let $\lim\limits_{i\rightarrow\infty}\Theta^{i}=\Theta^{\infty}$ and $\lim\limits_{i\rightarrow\infty}\Theta_{K_{x}}^{i}=\Theta_{K_{x}}^{\infty}$. Based on the above results regarding $K_{x}^{i}$ and $\Theta_{K_{x}}^{i}$, it holds that $\Theta^{i}_{K_{x}}\geq\Theta^{i+1}_{K_{x}}\geq\Theta^{\infty}_{K_{x}}$ [23]. Due to the full row rank property of $\bar{F}$, it follows that $\Theta^{i}\geq\Theta^{i+1}\geq\Theta^{\infty}$. From this non-increasing relation, $\Theta^{\infty}$ is a fixed point of equation (28). By the uniqueness of the optimal solution to the Bellman optimality equation, we have $\Theta^{\infty}=\Theta^{*}$ as the optimal solution. ∎

Lemma 4.3** (Second-order convergence rate).**

Algorithm 1 has a second-order convergence rate, that is,

[TABLE]

where $\beta_{1}:=\frac{\lambda_{\min}^{-1}(B^{\top}P^{K_{x}}B+R)\sigma^{2}_{\max}(\bar{F})\|\Theta^{0}\|}{\min\{\lambda_{\min}(Q_{x}),\lambda_{\min}(R)\}\sigma^{4}_{\min}(\bar{F})}$. For any $\epsilon>0$, when the iteration number satisfies $i\geq\log_{2}\left(\frac{\log(\beta_{1}\epsilon)}{\log(\beta\|\Theta^{0}-\Theta^{*}\|)}\right)$, Algorithm 1 yields an approximate optimal solution, i.e., $\|\Theta^{i}-\Theta^{*}\|\leq\epsilon$.

Proof.

Here, we focus on how the parameterization matrix $F$ affects the convergence rate parameter $\beta_{1}$, which is an important factor that demonstrates the advantage of Algorithm 1 over other PI algorithms.

Define $\bar{\Theta}_{K_{x}}^{*}:=\mathrm{diag}(\mathbf{0},\Theta_{K_{x},uu}^{-1})$, where $\Theta_{K_{x},uu}$ is the block matrix corresponding to the input of $\Theta_{K_{x}}$. Based on the controllers equivalence analyzed above and Lemma 4.2, equation $\Theta^{i+1}_{K_{x}}-\Theta^{*}_{K_{x}}=\sum_{j=1}^{\infty}((K_{AB}^{i})^{j})^{\top}(\Theta^{i}_{K_{x}}-\Theta^{*}_{K_{x}})\bar{\Theta}_{K_{x}}^{*}(\Theta^{i}_{K_{x}}-\Theta^{*}_{K_{x}})(K_{AB}^{i})^{j}$ in [23] still holds. Due to $\Theta_{K_{x},uu}=B^{\top}P^{K_{x}}B+R=\Theta_{uu}$, then $\bar{\Theta}_{K_{x}}^{*}=\mathrm{diag}(\mathbf{0},\Theta_{uu}^{-1})$. Combined further with the full row rank property of $\bar{F}$, the following equation under output feedback can be obtained.

[TABLE]

Taking the norm on both sides of the above equation yields

[TABLE]

where the second inequality follows from the fact that $K_{AB}^{i}$ satisfies a Lyapunov equation similar to (31), and for a symmetric positive definite matrix, the norm of its inverse equals the reciprocal of its minimum eigenvalue; the last inequality holds because Lemma 4.2 implies that $\{\Theta^{i}\}$ is monotonically decreasing. Rearranging the final inequality yields

[TABLE]

Simplifying it gives that when $i\geq\log_{2}\left(\frac{\log(\beta_{1}\epsilon)}{\log(\beta\|\Theta^{0}-\Theta^{*}\|)}\right)$, $\|\Theta^{i}-\Theta^{*}\|\leq\epsilon$ holds. ∎

4.2 Q-Value Based Value Iteration

This subsection focuses on an off-policy VI algorithm that does not require an initial stabilizing controller. If the parameter matrix $P$ of the current V-function is given, the following equation holds and contains only the unknown parameter $\Theta$ corresponding to the Q-function.

[TABLE]

Similar to the perspective of considering the system of equations in Subsection 4.1, here we focus on

[TABLE]

Due to the full row rank property of $\Psi_{0}$, we can compute $\Theta$ from equation (32). Note that under the greedy policy (25), the updated $P^{up}$ can be expressed as equation (26). Thus, we can use equations (32) and (26) as the Q-value evaluation and value update steps in VI, respectively. If $P$ has converged to the fixed point, the corresponding $\Theta$ can be used to obtain the optimal feedback gain via equation (25). The complete procedure is given in Algorithm 2.

Similarly to the analysis in Subsection 4.1, based on $X_{0}=FV_{0}$, $K_{x}=KF^{\dagger}$, $\Theta=\bar{F}^{\top}\Theta_{K_{x}}\bar{F}$ and $P=F^{\top}P^{K_{x}}F$, equation (32) can be transformed into

[TABLE]

From the full row rank property of $[X_{0}^{\top},U_{0}^{\top}]^{\top}$, equation (34) is equivalent to

[TABLE]

Because system (1) is free from noise, $X_{1}([X_{0}^{\top},U_{0}^{\top}]^{\top})^{\dagger}$ corresponds to the matrix parameters $[A,B]$ of system (1). Combining equation (35) with equation (26) shows that the iteration essence of Algorithm 2 is equivalent to the iteration of DARE (6). Therefore, Algorithm 2 is guaranteed to converge to the fixed point $P^{*}$, yielding an approximate optimal solution to Problem 3.13. Here, we will not provide a detailed proof of the convergence and optimality of Algorithm 2. Similarly, we focus more on how the parameterization matrix $F$ affects the convergence rate of Algorithm 2.

Lemma 4.4** (Linear convergence rate).**

Algorithm 2 has a linear convergence rate, that is,

[TABLE]

where $\beta_{2}:=\alpha(\|A+BK^{*}_{x}\|\frac{\sigma_{\max}(F)}{\sigma_{\min}(F)})^{2}(1+\sigma_{\min}^{-2}(F))$, with $\alpha$ being a constant dependent solely on the system (1) itself. For any $\epsilon>0$, when the iteration number satisfies $i\geq\frac{\log(\epsilon/\|P^{0}-P^{*}\|)}{\log\beta_{2}}$, Algorithm 2 yields an approximate optimal solution, i.e., $\|P^{i}-P^{*}\|\leq\epsilon$.

Proof.

Based on the above analysis of equations (35) and (26), the relation $P=F^{\top}P^{K_{x}}F$, and the full row rank property of $F$, if the influence of $F$ is considered in each step of the exact model-based convergence analysis in [22], the following equations can be derived under output feedback. When $P^{0}-P^{*}\succeq\mathbf{0}$,

[TABLE]

holds. When $P^{0}-P^{*}\preceq\mathbf{0}$ and $\|P^{i}-P^{*}\|\leq\frac{1}{\|BR^{-1}B^{\top}\|}$,

[TABLE]

holds. When $P^{0}-P^{*}\preceq\mathbf{0}$ and $\|P^{i}-P^{*}\|>\frac{1}{\|BR^{-1}B^{\top}\|}$, there exists a minimal integer $i_{0}$ such that $\|P^{i_{0}}-P^{*}\|\leq\frac{1}{\|BR^{-1}B^{\top}\|}$, then equation (36) still holds for $i\geq i_{0}$, while for $i<i_{0}$, the following inequality holds,

[TABLE]

When $P^{0}-P^{*}$ is indefinite, this case is a mixture of the aforementioned cases. From this, the convergence rate affected by $F$ is derived. ∎

4.3 Why Algorithms are Data-Efficient?

Previous works such as [8] did not consider the controllability of consistent systems or the availability of data matrices, instead, they directly perform PI or VI using the data sequence $\{\zeta_{t}\}$ collected under the state parameterization $x_{t}=E\zeta_{t}$. Furthermore, in numerical implementation, these approaches typically leverage two identities $s^{\top}Xr=(r\otimes s)^{\top}\mathrm{vec}(X)$ and $s^{\top}Ys=\mathrm{vech}(ss^{\top})^{\top}\mathrm{vech}(Y)$ (with arbitrary vectors $s$, $r$ and matrices $X$, $Y\in\mathbb{S}$). These identities transform the Bellman iterative equations into a LS problem, whose regression matrix comprises data matrices corresponding to $\mathrm{vech}(\zeta_{t}\zeta_{t}^{\top})$, $\mathrm{vech}(\zeta_{t+1}\zeta_{t+1}^{\top})$, $(u_{t}\otimes\zeta_{t})^{\top}$ and $\mathrm{vech}(u_{t}u_{t}^{\top})$. To ensure the solvability of the LS problem, the regression matrix must have full row rank, which is an indispensable condition for algorithm convergence in many classical works [10, 8]. However, when system (1) is noise-free, the full row rank condition of the regression matrix is not easily satisfied, especially in MO problems where it is nearly impossible. As shown in Section 3, consistent systems of MO problems must be uncontrollable, and the data matrix $Z_{0}$ fails to have full row rank, i.e., $\operatorname{rank}(Z_{0})=n_{v}<n_{\zeta}$. Let $\zeta_{t}:=\Pi a_{t}$ with $\Pi\in\mathbb{R}^{n_{\zeta}\times n_{v}}$ as the orthogonal basis matrix of $Z_{0}$, then $\mathrm{vech}(\zeta_{t}\zeta_{t}^{\top})=\mathrm{vech}(\Pi a_{t}a_{t}^{\top}\Pi^{\top})=(D^{\top}D)^{-1}D^{\top}(\Pi\otimes\Pi)D\cdot\mathrm{vech}(a_{t}a_{t}^{\top})$ holds, where $a_{t}\in\mathbb{R}^{n_{v}}$ is a coefficient vector, $D$ satisfies $\mathrm{vec}(\cdot)=D\cdot\mathrm{vech}(\cdot)$. Consequently, $[\mathrm{vech}(\zeta_{0}\zeta_{0}^{\top}),\cdots,\mathrm{vech}(\zeta_{T-1}\zeta_{T-1}^{\top})]=D^{\top}(\Pi\otimes\Pi)D[\mathrm{vech}(a_{0}a_{0}^{\top}),\cdots,\mathrm{vech}(a_{T-1}a_{T-1}^{\top})]$, and its rank must be less than or equal to $\operatorname{rank}([\mathrm{vech}(a_{0}a_{0}^{\top}),\cdots,\mathrm{vech}(a_{T-1}a_{T-1}^{\top})])\leq n_{v}(n_{v}+1)/2<n_{\zeta}(n_{\zeta}+1)/2$. Therefore, the regression matrix fails to satisfy the full row rank condition, and PI or VI algorithms under $x_{t}=E\zeta_{t}$ may not converge.

For the algorithms proposed in this paper, since we construct the substitute state $v_{t}$ by projecting the consistent system onto its controllable subspace, the data matrix $V_{0}$ must have full row rank. Moreover, both the generalized Sylvester equation (29) and equation (33) can be solved directly, and we do not use the LS method to solve the iterative equations. Therefore, there is no need to consider the rank condition of the additional regression matrix; Algorithm 1 and Algorithm 2 are guaranteed to converge for both SO and MO problems.

Due to the projection operation, the dimension of the substitute state reduces from $n_{\zeta}$ to $n_{v}$. Consequently, the number of unknowns in $\Theta$ and $P$ decreases from $(n_{\zeta}+m)(n_{\zeta}+m+1)/2$ and $(n_{\zeta})(n_{\zeta}+1)/2$ to $(n_{v}+m)(n_{v}+m+1)/2$ and $(n_{v})(n_{v}+1)/2$, respectively, with $n_{\zeta}-n_{v}=(p-1)n$. Further, in terms of data volume requirement, LS-based algorithms require the number of equations to exceed the number of unknowns, i.e., $T\geq(n_{\zeta}+m)(n_{\zeta}+m+1)/2$. In contrast, the proposed algorithms only require the number of columns in $\Psi_{0}$ of equations (29) and (33) to be no less than the number of its rows, i.e., $T\geq n_{v}+m$. Thus, Algorithm 1 and Algorithm 2 are data-efficient.

Meanwhile, note that the convergence rates in Lemma 4.3 and Lemma 4.4 are influenced by $\sigma_{\min}(F)$, and $F=EP$ according to Lemma 3.9. Without loss of generality, let $P$ be a Gauss transformation matrix. Then $F$ is equivalent to a matrix composed of partial columns of $E$. Therefore, $\sigma_{\min}(F)=\sqrt{\lambda_{\min}(F^{\top}F)}\leq\sqrt{\lambda_{\min}(E^{\top}E)}=\sigma_{\min}(E)$, where the inequality holds because the positive semi-definite matrix $F^{\top}F$ is a principal submatrix of $E^{\top}E$, and its minimum eigenvalue cannot exceed that of the original matrix. Consequently, compared to PI and VI algorithms under $x_{t}=E\zeta_{t}$ without processing, Algorithm 1 and Algorithm 2 achieve faster convergence and better numerical stability.

5 Robustness Analysis and Further Discussions

This section presents the robustness of Algorithm 1 and Algorithm 2, along with further discussions of the proposed methods.

5.1 Robustness Analysis

The above sections assume that the input-output data are exact. This subsection considers inexact data in the presence of small process noises $\{w_{t}\}$ and measurement noises $\{e_{t}\}$, which satisfy the following equation and are unmeasurable.

[TABLE]

In this case, the state parameterization based on delayed input-output (10) becomes

[TABLE]

Since $[w_{[t-n,t-1]}^{\top},e_{[t-n,t-1]}^{\top}]^{\top}$ is unmeasurable, we can still use $\xi_{t}$ for calculation. Actually, $\xi_{t}$ no longer serves as a substitute for $x_{t}$, but rather for $x_{t}-M^{\prime}[w_{[t-n,t-1]}^{\top},e_{[t-n,t-1]}^{\top}]^{\top}$. Next, we analyze the noise-induced perturbation on data matrices. Matrices with the superscript ex denote corresponding noise-free matrices. Define $W_{0}:=[w_{0},\cdots,w_{T-1}]$, $E_{0}:=[e_{0},\cdots,e_{T-1}]$, $W^{\mathcal{H}}:=\mathcal{H}_{N+1}(w_{[t-N,T-1]})$, $W_{0}^{\mathcal{H}}:=\mathcal{H}_{N}(w_{[t-N,T-2]})$, $W_{1}^{\mathcal{H}}:=\mathcal{H}_{N}(w_{[t-N+1,T-1]})$, $E_{0}^{\mathcal{H}}:=\mathcal{H}_{N}(e_{[t-N,T-2]})$, $E_{1}^{\mathcal{H}}:=\mathcal{H}_{N}(e_{[t-N+1,T-1]})$. It follows that

[TABLE]

[TABLE]

[TABLE]

For the state parameterization based on input-output filtering, the state estimation system (15) remains unchanged, and the error system (16) becomes $\epsilon_{t+1}=(A-LC)\epsilon_{t}+w_{t}-Le_{t}$. Since $w_{t}$ and $e_{t}$ are unmeasurable, we can still use system (18) to generate the substitute observation error vector. Thus, the practically applicable form of state parameterization based on input-output filtering remains (17), and $\eta^{\epsilon}_{t}$ no longer serves as a substitute for $\epsilon_{t}$ in reality, but rather for $\epsilon_{t}-w_{t}+Le_{t}$. For the noise-induced perturbation on data matrices, we can still consider the state estimation system (15) and the error system (16), and the difference lies in the process of generating the substitute vector. Due to the noise perturbation $\epsilon_{t}^{y}:=y_{t}-y_{t}^{ex}=\sum_{i=0}^{t-1}CA^{i}Bw_{i}+v_{t}$ in $y_{t}$ of the consistent system (20), then $\eta_{t}^{y}=\mathcal{R}_{t}(I_{p}\otimes A_{s},I_{p}\otimes b_{s})(y_{[0,t-1]}^{ex}+\epsilon_{[0,t-1]}^{y})$ holds. Define $D^{y}_{[i,j]}:=[\mathcal{R}_{i}(I_{p}\otimes A_{s},I_{p}\otimes b_{s})\epsilon_{[0,i]}^{y},\cdots,\mathcal{R}_{j}(I_{p}\otimes A_{s},I_{p}\otimes b_{s})\epsilon_{[0,j]}^{y}]$, it follows that

[TABLE]

By the norm property ofToeplitz matrix, it follows that $\|\mathcal{T}(A,I_{n},C)\|\leq\sum_{i=0}^{n-2}\|CA^{i}\|$. Therefore, the upper bound of $\|\Xi_{0}-\Xi_{0}^{ex}\|$ is $\sum_{i=0}^{n-2}\|CA^{i}\|\|W_{0}^{\mathcal{H}}\|+\|E_{0}^{\mathcal{H}}\|$. According to the form of $H_{0}-H_{0}^{ex}$, the upper bound of $\|H_{0}-H_{0}^{ex}\|$ is $C_{1}(\sum_{i=0}^{n-2}\|CA^{i}\|\|W_{0}^{\mathcal{H}}\|+\|E_{0}^{\mathcal{H}}\|)$ with a finite positive constant $C_{1}$. Consequently, the upper bound of $\|Z_{0}-Z_{0}^{ex}\|$ is $C_{2}(\sum_{i=0}^{n-2}\|CA^{i}\|\|W_{0}^{\mathcal{H}}\|+\|E_{0}^{\mathcal{H}}\|)$ with $C_{2}=\max\{1,C_{1}\}$. Thus, the upper bounds for the perturbation of data matrices are derived.

Consider the row rank of data matrix $Z_{0}$. Due to the presence of noises, $Z_{0}$ readily has full row rank, allowing it to be used directly in off-policy PI or VI. However, Lemma 3.4 and Lemma 3.8 are not entirely irrelevant; they provide critical guidance for mitigating noise impact, and this is another advantage of the proposed algorithms over other works. Note that $\sigma_{min}(Z_{0})$ may be extremely small, as these small singular values reflect noise influence. For this case, we can perform singular value decomposition (SVD) on the output-related submatrix of $Z_{0}$, and retain only components corresponding to the first $n$ largest singular values in absolute value. The selection of the number of singular values as $n$ is based on the ranks of controllability matrices derived in Lemma 3.4 and Lemma 3.8. This operation reduces the noise effect on $Z_{0}$ while preserving the true information of system (1). The resulting matrix $Z_{0}^{n}$ after SVD must have rank $n_{v}$. Furthermore, we can use Lemma 3.9 for projection to obtain the full row rank matrix $V_{0}$. These procedures not only reduce noise-induced perturbation of the data matrix but also enhance numerical stability.

Define $\Delta_{0}:=V_{0}-V_{0}^{ex}$ and $\Delta_{1}:=V_{1}-V_{1}^{ex}$. In addition to considering the case where noises increase the rank of $Z_{0}$, we must also ensure that noises are not too large to reduce the rank of $V_{0}$ or cause $V_{0}$ to have extremely small singular values. Therefore, the perturbation matrix needs to satisfy $\|\Delta_{0}\|<\sigma_{\min}(V_{0}^{ex})$ to prevent $V_{0}$ from being rank deficient.

The robustness of Algorithm 1 is now analyzed. Let $X_{0}=FV_{0}$, where $V_{0}$ is the noise-affected data matrix. However, $X_{1}\neq[A,B][I_{n},K_{x}^{\top}]^{\top}$, thus the state-dependent iterative formula derived from equation (28) should have an additional term $\Delta_{\epsilon}$ added to the right-hand side of equation (30), where

[TABLE]

Therefore, $[X_{0}^{\top},U_{0}^{\top}]^{\dagger}\Delta_{\epsilon}([X_{0}^{\top},U_{0}^{\top}]^{\top})^{\dagger}$ corresponds to a perturbation added to $\mathrm{diag}(Q_{x},R)$. When $\|\Delta_{\epsilon}\|<\sigma_{\min}(\mathrm{diag}(Q_{x},R))\sigma_{\min}^{2}(\bar{F}[V_{0}^{\top},U_{0}^{\top}]^{\top})$, the state-dependent iterative formula derived from (28) becomes a Lyapunov equation. Consequently, the stability and convergence of Algorithm 1 can be established analogously to the derivation in Lemma 4.2. Moreover, since equation (30) holds with $X_{0}^{ex}$ and $\Theta_{K_{x}}^{ex}$ replacing $X_{0}$ and $\Theta_{K_{x}}$, $\Delta\Theta_{K_{x}}:=\Theta_{K_{x}}-\Theta_{K_{x}}^{ex}$ satisfies

[TABLE]

By transforming $X_{0}$, $\Theta_{K_{x}}$ and $K_{x}$ in the above equations to their counterparts $V_{0}$, $\Theta$ and $K$ under output feedback, the following lemma can be obtained.

Lemma 5.1**.**

For system (37) with small process noises $\{w_{t}\}$ and measurement noise $\{v_{t}\}$, if $\|\Delta_{0}\|<\sigma_{\min}(V_{0}^{ex})$ and $\|\Delta_{\epsilon}\|<\sigma_{\min}(\mathrm{diag}(Q_{x},R))\sigma_{\min}^{2}(\bar{F}[V_{0}^{\top},U_{0}^{\top}]^{\top})$, then the sequence of output feedback gains $\{K^{i}\}$ generated by Algorithm 1 is stable, and $\{\Theta^{i}\}$ is monotonically decreasing and convergent.

Similarly, when analyzing the robustness of Algorithm 2, the state-dependent iterative equation derived from (28) remains (34). Moreover, the similar equation corresponding to exact matrices $\Theta^{ex}_{K_{x}}$, $X_{0}^{ex}$ and $X_{1}^{ex}$ also takes the same form, and by left-multiplying this equation by $[X_{0}^{ex\top},U_{0}^{\top}]^{\dagger}[X_{0}^{\top},U_{0}^{\top}]$ and right-multiplying it by $([X_{0}^{ex\top},U_{0}^{\top}]^{\dagger})^{\top}[X_{0}^{\top},U_{0}^{\top}]^{\top}$, we have

[TABLE]

Since $X_{1}^{ex}[X_{0}^{ex},U_{0}]^{\dagger}$ can be regarded as $[A,B]$, subtracting the above equation from equation (34) and using the definition $\Delta\Theta:=\Theta-\Theta^{ex}=\bar{F}^{\top}\Delta\Theta_{K_{x}}\bar{F}$, it holds that

[TABLE]

The perturbation manifests in the V-value update equation as

[TABLE]

Thus, the following lemma regarding robust convergence can be derived from Theorem 3 in [22].

Lemma 5.2**.**

For system (37) with small process noises $\{w_{t}\}$ and measurement noises $\{v_{t}\}$, if $\|\Delta_{0}\|<\sigma_{\min}(V_{0}^{ex})$ and $\|\Delta P\|<\sigma_{\min}(Q)$, Algorithm 2 converges.

5.2 Further Discussions

Owing to the state parameterization form adopted in this paper and the integration of Willems’ Fundamental Lemma into the algorithms, the proposed framework also offers the following advantages in flexibility and expandability.

• •

Note that the dimension $n$ of the true state $x_{t}$ is required for constructing the substitute state matrix $V_{0}$, and $x_{t}$ is unmeasurable. However, from equation (14), the rank of data matrix $[\mathcal{H}_{N}(u_{[0,T-1]})^{\top},\mathcal{H}_{N}(y_{[0,T-1]})^{\top}]^{\top}$ equals $mN+pN$ when $pN<n$, and $mN+n$ when $pN\geq n$. Thus, by incrementally increasing the positive integer $N$ from small to large and computing $\operatorname{rank}\left([\mathcal{H}_{N}(u_{[0,T-1]})^{\top},\mathcal{H}_{N}(y_{[0,T-1]})^{\top}]^{\top}\right)-mN$, the value of $n$ can be determined as the value that levels off with further increases in $N$.

• •

Benefiting from the properties of Lemma 2.5, the PE data required in this paper do not need to be collected continuously. If the mosaic Hankel matrix composed of multiple short trajectories satisfies the corresponding full row rank property, i.e., collective persistency of excitation [34], it also contains all the information of system (1) and can be equally applied to the proposed algorithms. This advantage is particularly useful for scenarios with data loss or unstable systems that only allow short experiments.

• •

Note that the controllability of the DT-LTI system is a sufficient condition in Lemma 2.5. In fact, without considering the controllability of system (1), Algorithm 1 and Algorithm 2 can still converge to the optimal solution as long as the full row rank data matrix $V_{0}$ is obtainable. The requirement for the controllability of system (1) in Section 3 is primarily for the convenience of theoretical analysis. Example 6.3 in Section 6 experimentally validates the applicability of proposed algorithms to uncontrollable systems.

• •

For DT problems, state parameterization (10) is more straightforward to implement than (17). However, this paper considers the state parameterization (17) not only because it represents a generalization of the former, but also because equation (10) is a dead-beat approach confined to DT scenarios, whereas state parameterization (17) can be naturally extended to continuous-time problems by replacing the z-transform with the Laplace transform. Thus, we expect that the proposed data utilization framework based on state parameterization (17) will provide some insights into efficiently solving continuous-time output feedback LQR problems.

• •

Furthermore, the state parameterization results in this paper may also facilitate solving output feedback LQR using policy optimization methods. By leveraging the capability of $[V_{0}^{\top},U_{0}^{\top}]^{\top}$ to parameterize the output feedback gain $K$, and the equivalence between $u_{t}=Kv_{t}$ and $u_{t}=K_{x}x_{t}$, the data-enabled policy optimization algorithm for state feedback [20] can be more generally extended to output feedback. However, the influence of the initial stabilizing controller should be noted.

6 Numerical Experiments

This section presents numerical experiments to validate four aspects of this work: the effectiveness of the method for calculating the state dimension $n$, the computational advantages of Algorithm 1 and Algorithm 2 in SO problems, the applicability of Algorithm 1 and Algorithm 2 to MO problems, and the efficacy of combining SVD with the controllability of consistent systems in mitigating noise effects.

Example 6.1** (Calculate $n$).**

Randomly generate a controllable DT-LTI system with $n=5$, $m=2$ and $p=2$. Use a random sequence as the input to this system, and collect the corresponding output data. For $N=1,2,\cdots$, calculate $\operatorname{rank}([\mathcal{H}_{N}(u_{[0,T-1]})^{\top},\mathcal{H}_{N}(y_{[0,T-1]})^{\top}]^{\top})-mN$; its variation is shown in Figure 2. This quantity stabilizes at $5$, which equals the state dimension $n$, thereby verifying the correctness of the method proposed in Subsection 5.2 for calculating $n$.

Example 6.2** (SO comparison problems).**

First, consider the third-order single-input single-output (SISO) aircraft system in [8], with system matrices given by

[TABLE]

[TABLE]

The initial state $x_{0}$, the eigenvalues of $A-LC$, $A_{\epsilon}$ and the initial state $\eta_{0}^{\epsilon}$ of system (18) are chosen identical to those in [8]. We initialize the random number generator in MATLAB using the rng(3) function. The excitation input to the system is selected as $u_{t}=\sum_{i=1}^{100}c_{i}\sin(a_{i}t+b_{i})$, where $a_{i}$, $b_{i}$ are generated from a uniform random distribution over $(0,2\pi)$, and $c_{i}$ is generated from a uniform random distribution over $(0,1)$. Available data are obtained under this setup. Similarly, the initial stabilizing controller for Algorithm 1 is set as $K^{0}=\mathbf{0}$, the initial matrix parameter for Algorithm 2 is set as $P^{0}=10^{5}I_{9}$, and the small threshold precision for both algorithms is set as $\epsilon=1$. The optimal output feedback gain obtained via Algorithm 1 is

[TABLE]

It is very close to the true optimal output feedback gain given in [8]. Compared with the LS-based PI algorithm in [8], which requires $5$ iterations and a running time of $0.017086$ seconds, Algorithm 1 only needs $4$ iterations to reduce the control gain difference $\|K^{i+1}-K^{i}\|$ to $9.2385\times 10^{-5}$, with a running time of $0.004024$ seconds. The optimal output feedback gain obtained via Algorithm 2 is

[TABLE]

Compared with the LS-based VI algorithm in [8], which requires $217$ iterations and a running time of $0.363406$ seconds, Algorithm 2 requires $185$ iterations to reduce the matrix parameter difference $\|P^{i+1}-P^{i}\|$ to $0.95764$, with a running time of $0.033872$ seconds. Thus, Algorithm 1 and Algorithm 2 are indeed more efficient.

Next, randomly generate 50 third-order SISO-LTI systems with $Q=2$, $R=1$. Choose the eigenvalues of $A-LC$ as $[-0.7,0.6,0.8]^{\top}$. $A_{\epsilon}$ is taken as the controllable canonical form of $A-LC$. We initialize the random number generator in MATLAB using the rng(3) function. $x_{0}$ and $\eta_{0}^{\epsilon}$ follow uniform distributions over $[-0.5,0.5]$. The excitation input is selected as $u_{t}=\sum_{i=1}^{100}c_{i}\sin(a_{i}t+b_{i})$, where $a_{i}$, $b_{i}$ are generated from a uniform random distribution over $(0,2\pi)$ and $c_{i}$ is generated from a uniform random distribution over $(0,1)$. Available data are obtained under this setup. Set $K^{0}=\mathbf{0}$ for PI algorithms, $P^{0}=10^{3}I_{9}$ for VI algorithms, and $\epsilon=0.01$. Note that LS-based algorithms in [8] require regression matrices to have full row rank (where the full row rank property of the PI regression matrix can be derived from that of the VI regression matrix), while Algorithm 1 and Algorithm 2 focus on the full row rank property of $V_{0}$. Therefore, we first compare the minimum absolute singular value of the regression matrix in the LS-based VI algorithm and that of $V_{0}$, as shown in Figure 3.

All the orders of magnitude corresponding to LS-based VI regression matrices are less than or equal to $10^{-7}$, whereas only $1$ value among those corresponding to $V_{0}$ have an order of magnitude less than $10^{-7}$. This indicates that regression matrices of LS-based algorithms may be closer to singularity, leading to slow convergence or even non-convergence of algorithms, and causing numerical instability issues. In contrast, the proposed Algorithm 1 and Algorithm 2 alleviate this problem to a certain extent. The average running times and average numbers of iterations for LS-based algorithms, Algorithm 1, and Algorithm 2 are shown in Table 1 and Table 2. These results further verify the complete data parameterization capability and high efficiency of the data matrix under the state parameterization constructed in Subsection 4.3.

Example 6.3** (MO problems).**

Consider the controllable and observable LTI system with $n=4$, $m=2$ and $p=2$, where

[TABLE]

Choose the eigenvalues of $A-LC$ as $[0.8994,-0.6,0.7,0]^{\top}$. $A_{\epsilon}$ is taken as the controllable canonical form of $A-LC$. We initialize the random number generator in MATLAB using the rng(1) function. $x_{0}$ and $\eta_{0}^{\epsilon}$ follow uniform distributions over $[-0.5,0.5]$. The excitation input is set as $u_{t}=\sum_{i=1}^{100}c_{i}\sin(a_{i}t+b_{i})$, where $a_{i}$, $b_{i}$ are generated from a uniform random distribution over $(0,2\pi)$ and $c_{i}$ is generated from a uniform random distribution over $(0,1)$. Available data are obtained under this setup. Then $n_{\zeta}=20$, $n_{v}=18$, and $Z_{0}$ does not have full row rank. The rank of the LS-based VI regression matrix in [8] is $152$, which is smaller than its number of rows $253$, thus the corresponding LS-based algorithms may not converge. Furthermore, for algorithms in [8], the number of unknowns is $253$, and the required data amount is $T\geq 253$. By contrast, for Algorithm 1 and Algorithm 2, the projected $V_{0}$ must have full row rank, with the number of unknowns being $210$ and the required data amount $T\geq 20$.

The small threshold precision of algorithms is set as $\epsilon=0.001$. For Algorithm 1, set $K^{0}=\mathbf{0}$; the convergence and the input-output trajectories under the resulting output feedback gain $K_{PI}^{*}$ are shown in Figure 4 and Figure 5. Algorithm 1 achieves a residual of $2.7537\times 10^{-6}$ after $5$ iterations. Noting the true optimal cost $x_{0}^{\top}P^{K_{x}^{*}}x_{0}=0.5506$ and the calculated optimal cost $\eta_{0}^{\top}[I_{n_{v}},K_{PI}^{*\top}]\Theta^{K_{PI}^{*}}[I_{n_{v}},K_{PI}^{*\top}]^{\top}\eta_{0}=0.5506$, with a difference of $-3.9426\times 10^{-7}$, $K_{PI}^{*}$ is thus a near-optimal solution.

For Algorithm 2, set $P^{0}=10^{3}I_{18}$; the convergence and the input-output trajectories under the resulting output feedback gain $K_{VI}^{*}$ are shown in Figure 6 and Figure 7. Algorithm 2 achieves a residual of $9.9408\times 10^{-5}$ after $105$ iterations. Noting the true optimal cost $x_{0}^{\top}P^{K_{x}^{*}}x_{0}=0.5506$ and the calculated optimal cost $\eta_{0}^{\top}[I_{n_{v}},K_{VI}^{*\top}]\Theta^{K_{VI}^{*}}[I_{n_{v}},K_{VI}^{*\top}]^{\top}\eta_{0}=0.5506$ with a difference of $-5.0548\times 10^{-8}$, $K_{VI}^{*}$ is thus a near-optimal solution.

Next, consider the uncontrollable but stabilizable and observable LTI system with $n=4$, $m=2$ and $p=2$, where

[TABLE]

The controllability matrix of this system has a rank of $3$. The remaining settings are the same as those in the aforementioned controllable MO example. This setup ensures that $F$ and $V_{0}$ in this example have full row rank; thus, Algorithm 1 and Algorithm 2 are feasible even if this system is uncontrollable. Algorithm 1 achieves a residual of $2.7834\times 10^{-4}$ after $3$ iterations. Noting the true optimal cost $x_{0}^{\top}P^{K_{x}^{*}}x_{0}=0.1468$ and the calculated optimal cost $\eta_{0}^{\top}[I_{n_{v}},K_{PI}^{*\top}]\Theta^{K_{PI}^{*}}[I_{n_{v}},K_{PI}^{*\top}]^{\top}\eta_{0}=0.1468$ with a difference of $2.1284\times 10^{-10}$, $K_{PI}^{*}$ is thus a near-optimal solution. Algorithm 2 achieves a residual of $9.2075\times 10^{-5}$ after $68$ iterations. Noting the true optimal cost $x_{0}^{\top}P^{K_{x}^{*}}x_{0}=0.1468$ and the calculated optimal cost $\eta_{0}^{\top}[I_{n_{v}},K_{VI}^{*\top}]\Theta^{K_{VI}^{*}}[I_{n_{v}},K_{VI}^{*\top}]^{\top}\eta_{0}=0.1468$ with a difference of $-4.3531\times 10^{-6}$, $K_{VI}^{*}$ is thus a near-optimal solution. These results validate the conclusion regarding controllability presented in Subsection 5.2.

Example 6.4** (Robustness Problem).**

Consider the same system (38) as in the controllable MO problem in Example 6.3 with identical user-defined parameters. Let the absolute upper bounds of noises $\{w_{t}\}$ and $\{e_{t}\}$ both be $W_{\max}$. When $W_{\max}=10^{-4}$, the data matrix $Z_{0}$ has full row rank. If $100$ trials are conducted to construct the regression matrices of LS-based VI in [8], the average row rank is $221.58$, and none of them achieves full row rank. Thus, under the influence of noise, LS-based PI and VI algorithms also barely satisfy the feasibility conditions. In this example, we initialize the random number generator in MATLAB using the rng(75) function. If Algorithm 1 is directly run with $Z_{0}$, a residual of $4.5349\times 10^{-4}$ is achieved after $4$ iterations. Noting the true optimal cost $x_{0}^{\top}P^{K_{x}^{*}}x_{0}=1.9109$ and the calculated optimal cost $\eta_{0}^{\top}[I_{n_{v}},K_{PI}^{*\top}]\Theta^{K_{PI}^{*}}[I_{n_{v}},K_{PI}^{*\top}]^{\top}\eta_{0}=1.9550$, the difference is $-0.0441$. If Algorithm 1 is run with $V_{0}$ (constructed via SVD and projection as described in Subsection 5.1), a residual of $6.7282\times 10^{-4}$ is achieved after $4$ iterations. Noting the calculated optimal cost $\eta_{0}^{\top}[I_{n_{v}},K_{PI}^{*\top}]\Theta^{K_{PI}^{*}}[I_{n_{v}},K_{PI}^{*\top}]^{\top}\eta_{0}=1.9115$, the difference is $-6.1031\times 10^{-4}$ from $x_{0}^{\top}P^{K_{x}^{*}}x_{0}$, which is a clear improvement. The convergence of Algorithm 1 corresponding to $Z_{0}$ and $V_{0}$ is shown in Figure 8. If Algorithm 2 is directly run with $Z_{0}$, it may fail to converge. However, when run with $V_{0}$, Algorithm 2 converges.

Next, set $W_{\max}$ to $10^{-3}$, $10^{-4}$ and $10^{-6}$, respectively. Let Algorithm 1 with $V_{0}$ be run for $6$ iterations, and Algorithm 2 with $V_{0}$ be run $150$ iterations. Define $\Delta_{K}$ as the value of $\|K^{i+1}-K^{i}\|$ and $\Delta_{xPx}$ as the value of $x_{0}^{\top}P^{K_{x}^{*}}x_{0}-\eta_{0}^{\top}[I_{n_{v}},K^{i+1\top}]\Theta^{i+1}[I_{n_{v}},K^{i+1\top}]^{\top}\eta_{0}$ obtained at the final iteration. The error results are shown in Table 3 and Table 4. When $W_{\max}=10^{-3}$, the results generated by Algorithm 1 and Algorithm 2 have a certain small gap from those under exact data. However, when $W_{\max}$ is small, the results of Algorithm 1 and Algorithm 2 are very close to those under exact data.

7 Conclusion

This work develops a general and efficient data-driven framework for designing the optimal output feedback controllers for LQR problems in a model-free manner. Specifically, we focus on state parameterization, providing solid theoretical foundations for the complete data-parameterized closed-loop systems and output feedback controllers. The proposed algorithms fully exploit the inherent information and structure of data, exhibiting distinct advantages in data usage conditions and data volume requirements. This framework is amenable to extension to various optimal control problems; meanwhile, the data parameterization theory presented in this paper can be deeply integrated with multiple data-driven approaches, offering new insights for research in related fields. However, it should be noted that state parameterization is essentially a redundant representation of unmeasurable states, and its application to high-dimensional systems may lead to increased computational complexity. Therefore, how to achieve model-free optimal output feedback control for LQR problems in high-dimensional scenarios remains a key research topic to be further explored in the future. Additionally, extending the proposed algorithms to non-restricted small-noise cases is also one of our core research directions for subsequent work.