## 2 Output Model Reference Control for Unknown Systems  
Let the discrete-time nonlinear unknown open-loop minimum-phase state-space deterministic strictly causal process be  
\[P:\left\{ \begin{array}{l}\mathbf{x}_{k+1}=\mathbf{f}(\mathbf{x}_{k},\mathbf{ u}_{k}),\\ \mathbf{y}_{k}=\mathbf{g}(\mathbf{x}_{k}),\end{array} \right. \tag{1}\]  
where \(k\) indexes the discrete time, \(\mathbf{x}_{k}=\left[x_{k,1}\ \ldots\ x_{k,n}\right]^{T}\in X\subset\Re^{n}\) is the \(n\)-dimensional state vector (upper \(T\) is matrix transpose), \(\mathbf{u}_{k}=\left[u_{k,1},\ldots,u_{k,m}\right]^{T}\in U\subset\Re^{m}\) is the control input signal, \(\mathbf{y}_{k}=\left[y_{k,1},\ldots,y_{k,p}\right]^{T}\in Y\subset\Re^{p}\) is the measurable controlled output, \(\mathbf{f}:X\times U\to X\) is an unknown nonlinear system function, \(\mathbf{g}:X\to Y\) is an unknown nonlinear output function of the states, and the initial conditions are not considered for analysis at this point. It is further assumed that the definition domains \(X\), \(U\), \(Y\) are compact convex. The following assumptions common to the data-driven problem formulation [1] are:  
_A1_: System (1) is controllable and fully state observable. _A2_: System (1) is internally stable on \(X\times U\). Assumptions _A1_ and _A2_ are common in the data-driven control literature and difficult to assess when unknown process models are assumed. They may be supported from the experience on the process operation or from the literature. If no knowledge exists whatsoever, control can be tried in the constraining domains related to the minimum safety operating conditions of the process, which is required minimum information on the process variables. Internal stability is sufficient for output feedback control design and necessary for state-feedback control design using input-state samples. Concerning the controllability and full state observability assumption _A1_ imposed to the process, if the observability cannot be verified analytically, data-driven observers can be built using past samples of either the inputs and outputs and/or of the partially measurable state, as shown for linear systems in [33; 34] and used for nonlinear systems in [35]. State measurement requires more insight on the process than several pure input-output representations. Equation (1) is a general form for most controlled processes in practice and it is not restrictive. In this form, it obeys the definition of a deterministic Markov decision process. The discrete-time known open-loop stable minimum-phase state-space deterministic strictly causal ORM is  
\[ORM:\left\{ \begin{array}{l}\mathbf{x}_{k+1}^{m}=\mathbf{f}^{m}(\mathbf{x}_{ k}^{m},\mathbf{r}_{k}),\\ \mathbf{y}_{k}^{m}=\mathbf{g}^{m}(\mathbf{x}_{k}^{m}),\end{array} \right. \tag{2}\]  
where \(\mathbf{x}_{k}^{m}=\left[x_{k,1}^{m},\ldots,x_{k,m}^{m}\right]^{T}\in X_{m} \subset\Re^{n_{m}}\) is the state vector of the ORM, \(\mathbf{r}_{k}=\left[r_{k,1},\ldots,r_{k,p}\right]^{T}\in R_{m}\subset\Re^{p}\) is the reference input signal, \(\mathbf{y}_{k}^{m}=\left[y_{k,1}^{m},\ldots,y_{k,p}^{m}\right]^{T}\in Y_{m} \subset\Re^{p}\) is the ORM's output, \(\mathbf{f}^{m}:X_{m}\times R_{m}\to X_{m}\), \(\mathbf{g}^{m}:X_{m}\to Y_{m}\) are known nonlinear maps. Initial conditions are zero unless stated otherwise. Note that \(\mathbf{r}_{k},\mathbf{y}_{k},\mathbf{y}_{k}^{m}\) have the same size \(p\) for square feedback control systems. If the ORM (2) is linear time-invariant in particular, it is always possible to express the ORM as an input-output linear time-invariant transfer matrix \(\mathbf{y}_{k}^{m}=\mathbf{M}(z)\mathbf{r}_{k}\), where \(\mathbf{M}(z)\) is an asymptotically stable unit gain (i.e., \(\mathbf{M}(1)=\mathbf{I}\), where \(\mathbf{I}\) is the identity matrix) rational transfer matrix and \(\mathbf{r}_{k}\) is the reference input that drives both the feedback control system and the ORM.