Data-driven control of nonlinear systems from input-output data*

A dataset consisting of open-loop input-output measurements

is available, where the positive integers $N,T$ will be specified later. The samples in the dataset are obtained from off-line experiment(s) conducted on system (1), hence they satisfy the equations (1), namely

For our purpose of designing an output feedback controller from $\mathcal{D}$ it is not required that all the samples of the dataset are sequentially obtained in a single experiment. In fact, even multiple experiments collecting $N+T$ samples suffice. This is useful especially when dealing with unstable dynamics.

The problem of interest is to design an output feedback controller that stabilizes the nonlinear system, based on the dataset $\mathcal{D}$. To this purpose, we need to infer the behavior of the state $x$ from input-output measurements, for which suitable “observability” conditions on the system (1) are required. Before stating them, we introduce some notation. We let

Note that (3) gives $x(k)=F^{N}(x(k-N),u_{[k-N,k-1]})$. To reduce the notational complexity, we introduce $v_{[0,k]}$, which denotes the sequence of values $v_{0},\ldots,v_{k}$. Hence, the last identity above is rewritten as $F^{k+1}(x,v_{[0,k]}):=f(F^{k}(x,v_{[0,k-1]}),v_{k})$. In what follows, we will use symbols like $v_{[0,k]}$ also to denote the vector $\left[\begin{smallmatrix}v_{0}&v_{1}&\ldots&v_{k}\end{smallmatrix}\right]^{\top}$.

The following is the main assumption on system (1).

Following [22, Definition 1], we refer to the assumption above as a uniform observability on $\mathcal{X}$ property. It is observed in [22] that, if $f,h$ are continuously differentiable functions, uniform observability is not restrictive in the sense that a nonuniform distinguishability property and a nonuniform observability rank condition imply uniform observability. Since for any $M\geq N$ the mapping $\Phi_{M}$ remains injective, we do not need to know the smallest $N$ for which Assumption 1 holds.

For any $v_{[0,N-2]}\in\mathcal{U}^{N-1}$, the function

such that $x\mapsto w=\Phi_{N}(x,v_{[0,N-2]})$, is injective on $\mathcal{X}$ and one can define a left inverse

such that $\Psi_{N}(\Phi_{N}(x,v_{[0,N-2]}),v_{[0,N-2]})=x$ for all $x\in\mathcal{X}$.

We introduce a system equivalent to (1) which is better suited for control design. By equivalent it is meant that the new system has the same input-output behavior of system (1) when properly initialized. We use this auxiliary system for control design purposes. Later on we show the effect of the designed controller on the actual system (1).

For any $v_{[0,N-1]}\in\mathbb{R}^{N}$, define the functions

with the pair $(A_{c},B_{c})\in\mathbb{R}^{N\times N}\times\mathbb{R}^{N}$ in the Brunovsky form. The domain of $\psi(\cdot,v_{[0,N-1]}),\tilde{h}(\cdot,v_{[0,N-1]})$, $\tilde{f}(\cdot,v_{[0,N-1]})$ is $\Phi_{N}(\mathcal{X},v_{[0,N-2]})$. Under the standing assumptions on $f,h$, these functions are continuous and zero at $(w,v)=(0,0)$.

In the result below, for a $k\in\mathbb{Z}$, we let $u_{[k-N,k-1]}$ be an input sequence applied to system (1) and $y_{[k-N,k-1]}$ its output response from some initial condition $x(k-N)$.

Proof. For the sake of completeness, it is given in Appendix VI-A.  $\blacksquare$

System (6) is driven by the past $N$ samples of $u$, which is the input to (1). These past values are obtained by adding a chain of integrators to the dynamics (6)

with the interconnection condition

which returns the system

Once the system’s state satisfies $(w(\overline{k}),\xi(\overline{k}))=(y_{[\overline{k}-n,\overline{k}-1]},u_{[\overline{k}-n,\overline{k}-1]})$ for some $\overline{k}\in\mathbb{Z}$, the input-output behavior of this system matches the one of (1) for all $k\geq\overline{k}$. We will discuss later on the availability of such initial condition at a time $\overline{k}$.

To obtain $u$ that drives the chain of integrators making the dynamic controller, we argue as in [2, 15]. We first introduce the following:

This is a technical assumption due to the need to give the nonlinearities of (10) a form for which the controller design is possible. Although it is restrictive, [15, Section VI.B] bypasses such an assumption by expressing $\tilde{h}(w,\xi)$ as $\alpha Z(w,\xi)+d(w,\xi)$, where the term $d(w,\xi)$ represents the nonlinearities that were excluded from $Z(w,\xi)$, and then analyzing the stability of the system in the presence of the neglected nonlinearity $d(w,\xi)$. This analysis goes beyond the scope of this paper.

We consider the case in which the function $Z(w,\xi)$ comprises both a linear part and a nonlinear part $Q(w,\xi)$, i.e.

The system (10) can then be written as

where

and the pair $(A_{c},B_{c})$ is in the Brunovsky canonical form.

We focus on the case in which the input $u$ is designed as a function of $Z(w,\xi)$, i.e.

where $\kappa\in\mathbb{R}^{1\times S}$ is the control gain. Write the closed-loop system (11)-(12) as

The system is defined for any $\xi\in\mathcal{U}^{N}$ and any $w\in\Phi_{N}(\mathcal{X},\xi_{[1,N-1]})$.

Preliminary to the design of the controller is a data-dependent representation of the closed-loop system. We first introduce some notation. Recall the dataset in (2) and introduce, for $i=0,\ldots,T-1$,

We assume that the samples of the dataset evolve in the domain of definition of (13).

We let:

In the definition of $Q_{0}$, we are using the shorthand notation $Q(i)$ for $Q(Y(i),U(i))$. Under Assumption 3, bearing in mind the dynamics (11), the dataset-dependent matrices introduced in (14) satisfy

We establish the following:

Proof. For any $\xi\in\mathcal{U}^{N}$ and any $w\in\Phi_{N}(\mathcal{X},\xi_{[1,N-1]})$, it holds

By (16), we obtain that

Hence,

$\blacksquare$

Let the set of real-valued symmetric matrices of dimension $n\times n$ be denoted by $\mathbb{S}^{n\times n}$. This data-dependent representation leads to the following local stabilization result:

Proof. Set $G_{1}=\mathcal{Y}_{1}\mathcal{P}_{1}^{-1}$. Then (18b), (18d) imply

which along with the definition of $\kappa$ in (22), namely $\kappa=U_{0}\left[\begin{smallmatrix}G_{1}&G_{2}\end{smallmatrix}\right]$, implies (16). Hence, the data-dependent representation of system (13) given in Lemma 2 holds. By Schur complement, the constraint (18c) is equivalent to

Pre- and post-multiplying by $\mathcal{P}_{1}^{-1}$ and bearing in mind the definition of $G_{1}$ we obtain

which shows that $V(w,\xi)=\begin{bmatrix}w^{\top}&\xi^{\top}\end{bmatrix}\mathcal{P}_{1}^{-1}\begin{bmatrix}w\\ \xi\end{bmatrix}$ is a Lyapunov function for the linear part of the closed-loop system. In particular note that the domain of definition of the function $V(w,\xi)$ is the same as the one of system (23), hence, $V(w,\xi)$ is defined at the origin. We have

In view of (19), $V(w^{+},\xi^{+})-V(w,\xi)<0$ in a neighborhood of the origin. This shows the claim.  $\blacksquare$

Proposition 1 provides a local stabilization result. Following [15], Proposition 1 can be extended to provide an estimate of the Region of Attraction (ROA) of the system (23). First we recall the following definitions.

Recall the Lyapunov difference

with $M,N$ as in (17).

As the function $\mathcal{W}(w,\xi)$ is known from the data, the estimate of the ROA $\mathcal{R}_{\gamma}$ is computable.