# Learning Output Reference Model Tracking for Higher-Order Nonlinear Systems with Unknown Dynamics +
## 2 Output Model Reference Control for Unknown Dynamics Nonlinear Processes  
### The Process  
A discrete-time nonlinear unknown open-loop stable state-space deterministic strictly causal process is defined as [12; 46]  
\[P:\{\mathbf{x}_{k+1}=\mathbf{f}(\mathbf{x}_{k},\mathbf{u}_{k}),\mathbf{y}_{k}= \mathbf{g}(\mathbf{x}_{k})\}, \tag{1}\]  
where \(k\) indexes the discrete time, \(\mathbf{x}_{k}=[x_{k,1},...,x_{k,n}]^{\top}\in\Omega_{X}\subset\mathbb{R}^{n}\) is the \(n\)-dimensional state vector, \(\mathbf{u}_{k}=[u_{k,1},...,u_{k,m_{k}}]^{\top}\in\Omega_{U}\subset\mathbb{R} ^{m_{u}}\) is the control input signal, \(\mathbf{y}_{k}=[y_{k,1},...,y_{k,p}]^{\top}\in\Omega_{Y}\subset\mathbb{R}^{p}\) is the measurable controlled output, \(\mathbf{f}:\Omega_{X}\times\Omega_{U}\mapsto\Omega_{X}\) is an _unknown_ nonlinear system function continuously differentiable within its domain, \(\mathbf{g}:\Omega_{X}\mapsto\Omega_{Y}\) is an unknown nonlinear continuously differentiable output function. Initial conditions are not accounted for at this point. Assume known domains \(\Omega_{X},\Omega_{U},\Omega_{Y}\) are compact convex. Equation (1) is a general un-restrictive form for most controlled processes. The following assumptions common to the data-driven formulation are [12; 46]:  
**Assumption 1** (A1).: _(1) is fully state controllable with measurable states._  
**Assumption 2** (A2).: _(1) is input-to-state stable on known domain \(\Omega_{U}\times\Omega_{X}\)._  
**Assumption 3** (A3).: _(1) is minimum-phase (MP)._  
A1 and A2 are widely used in data-driven control, cannot be checked analytically for the unknown model (1) but can be inferred from historical and working knowledge with the process. Should such information not be available, the user can attempt process control under restraining safety operating conditions, that are usually dealt with at supervisory level control. Input to state stability (A2) is necessary if open-loop input-state samples collection is intended to be used for state space control design. Assumption A2 can be relaxed if a stabilizing state-space controller is already available and used just for the purpose of input-state data collection. A3 is the least restrictive assumption and it is used in the context of the VRFT design of a feedback controller based on input-output process data. Although solutions exist to deal with nonminimum-phase systems processes, the MP assumption simplifies the VRFT design and the output reference model selection (to be introduced in the following section).  
_Comment 1_. [46] Model (1) accounts for a wide range of processes including fixed time-delay ones. For positive integer nonzero delay \(d\) on the control input \(\mathbf{u}_{k-d}\), additional states can extend the initial process model (1) as \(\mathbf{x}_{k,1}^{m}=\mathbf{u}_{k-1},\mathbf{x}_{k,2}^{u}=\mathbf{u}_{k-2},...,\mathbf{x}_{k,d}^{u}=\mathbf{u}_{k-d}\) and arrive at a state-space model without delays, in which the additional states are measurable as past input samples. A delay in the original states in (1), i.e., \(\mathbf{x}_{k-d}\), are similarly treated.