Data-driven control system

The standard approach to control systems design is organized in two-steps:

1. Model identification aims at estimating a nominal model of the system ${\displaystyle {\widehat {G}}=G\left(q;{\widehat {\theta }}_{N}\right)}$, where ${\displaystyle q}$ is the unit-delay operator (for discrete-time transfer functions representation) and ${\displaystyle {\widehat {\theta }}_{N}}$ is the vector of parameters of ${\displaystyle G}$ identified on a set of ${\displaystyle N}$ data. Then, validation consists in constructing the uncertainty set ${\displaystyle \Gamma }$ that contains the true system ${\displaystyle G_{0}}$ at a certain probability level.
2. Controller design aims at finding a controller ${\displaystyle C}$ achieving closed-loop stability and meeting the required performance with ${\displaystyle {\widehat {G}}}$.

Typical objectives of system identification are to have ${\displaystyle {\widehat {G}}}$ as close as possible to ${\displaystyle G_{0}}$, and to have ${\displaystyle \Gamma }$ as small as possible. However, from an identification for control perspective, what really matters is the performance achieved by the controller, not the intrinsic quality of the model.

One way to deal with uncertainty is to design a controller that has an acceptable performance with all models in ${\displaystyle \Gamma }$, including ${\displaystyle G_{0}}$. This is the main idea behind robust control design procedure, that aims at building frequency domain uncertainty descriptions of the process. However, being based on worst-case assumptions rather than on the idea of averaging out the noise, this approach typically leads to conservative uncertainty sets. Rather, data-driven techniques deal with uncertainty by working on experimental data, and avoiding excessive conservativism.

In the following, the main classifications of data-driven control systems are presented.

The iterative feedback tuning (IFT) method was introduced in 1994,[2] starting from the observation that, in identification for control, each iteration is based on the (wrong) certainty equivalence principle.

IFT is a model-free technique for the direct iterative optimization of the parameters of a fixed-order controller; such parameters can be successively updated using information coming from standard (closed-loop) system operation.

Let ${\displaystyle y^{d}}$ be a desired output to the reference signal ${\displaystyle r}$; the error between the achieved and desired response is ${\displaystyle {\tilde {y}}(\rho )=y(\rho )-y^{d}}$. The control design objective can be formulated as the minimization of the objective function:

${\displaystyle J(\rho )={\frac {1}{2N}}\sum _{t=1}^{N}E\left[{\tilde {y}}(t,\rho )^{2}\right].}$

Given the objective function to minimize, the quasi-Newton method can be applied, i.e. a gradient-based minimization using a gradient search of the type:

${\displaystyle \rho _{i+1}=\rho _{i}-\gamma _{i}R_{i}^{-1}{\frac {d{\widehat {J}}}{d\rho }}(\rho _{i}).}$

The value ${\displaystyle \gamma _{i}}$ is the step size, ${\displaystyle R_{i}}$ is an appropriate positive definite matrix and ${\displaystyle {\frac {d{\widehat {J}}}{d\rho }}}$ is an approximation of the gradient; the true value of the gradient is given by the following:

${\displaystyle {\frac {dJ}{d\rho }}(\rho )={\frac {1}{N}}\sum _{t=1}^{N}\left[{\tilde {y}}(t,\rho ){\frac {\delta y}{\delta \rho }}(t,\rho )\right].}$

The value of ${\displaystyle {\frac {\delta y}{\delta \rho }}(t,\rho )}$ is obtained through the following three-step methodology:

1. Normal Experiment: Perform an experiment on the closed loop system with ${\displaystyle C(\rho )}$ as controller and ${\displaystyle r}$ as reference; collect N measurements of the output ${\displaystyle y(\rho )}$, denoted as ${\displaystyle y^{(1)}(\rho )}$.
2. Gradient Experiment: Perform an experiment on the closed loop system with ${\displaystyle C(\rho )}$ as controller and 0 as reference ${\displaystyle r}$; inject the signal ${\displaystyle r-y^{(1)}(\rho )}$ such that it is summed to the control variable output by ${\displaystyle C(\rho )}$, going as input into the plant. Collect the output, denoted as ${\displaystyle y^{(2)}(\rho )}$.
3. Take the following as gradient approximation: ${\displaystyle {\frac {\delta {\widehat {y}}}{\delta \rho }}(\rho )={\frac {\delta C}{\delta \rho }}(\rho )y^{(2)}(\rho )}$.

A crucial factor for the convergence speed of the algorithm is the choice of ${\displaystyle R_{i}}$; when ${\displaystyle {\tilde {y}}}$ is small, a good choice is the approximation given by the Gauss–Newton direction:

${\displaystyle R_{i}={\frac {1}{N}}\sum _{t=1}^{N}{\frac {\delta {\widehat {y}}}{\delta \rho }}(\rho _{i}){\frac {\delta {\widehat {y}}^{T}}{\delta \rho }}(\rho _{i}).}$

Noniterative correlation-based tuning (nCbT) is a noniterative method for data-driven tuning of a fixed-structure controller.[3] It provides a one-shot method to directly synthesize a controller based on a single dataset.

Suppose that ${\displaystyle G}$ denotes an unknown LTI stable SISO plant, ${\displaystyle M}$ a user-defined reference model and ${\displaystyle F}$ a user-defined weighting function. An LTI fixed-order controller is indicated as ${\displaystyle K(\rho )=\beta ^{T}\rho }$, where ${\displaystyle \rho \in \mathbb {R} ^{n}}$, and ${\displaystyle \beta }$ is a vector of LTI basis functions. Finally, ${\displaystyle K^{*}}$ is an ideal LTI controller of any structure, guaranteeing a closed-loop function ${\displaystyle M}$ when applied to ${\displaystyle G}$.

The goal is to minimize the following objective function:

${\displaystyle J(\rho )=\left\|F{\bigg (}{\frac {K^{*}G-K(\rho )G}{(1+K^{*}G)^{2}}}{\bigg )}\right\|_{2}^{2}.}$

${\displaystyle J(\rho )}$ is a convex approximation of the objective function obtained from a model reference problem, supposing that ${\displaystyle {\frac {1}{(1+K(\rho )G)}}\approx {\frac {1}{(1+K^{*}G)}}}$.

When ${\displaystyle G}$ is stable and minimum-phase, the approximated model reference problem is equivalent to the minimization of the norm of ${\displaystyle \varepsilon (t)}$ in the scheme in figure.

The idea is that, when G is stable and minimum phase, the approximated model reference problem is equivalent to the minimization of the norm of ${\displaystyle \varepsilon }$.

The input signal ${\displaystyle r(t)}$ is supposed to be a persistently exciting input signal and ${\displaystyle v(t)}$ to be generated by a stable data-generation mechanism. The two signals are thus uncorrelated in an open-loop experiment; hence, the ideal error ${\displaystyle \varepsilon (t,\rho ^{*})}$ is uncorrelated with ${\displaystyle r(t)}$. The control objective thus consists in finding ${\displaystyle \rho }$ such that ${\displaystyle r(t)}$ and ${\displaystyle \varepsilon (t,\rho ^{*})}$ are uncorrelated.

The vector of instrumental variables ${\displaystyle \zeta (t)}$ is defined as:

${\displaystyle \zeta (t)=[r_{W}(t+\ell _{1}),r_{W}(t+\ell _{1}-1),\ldots ,r_{W}(t),\ldots ,r_{W}(t-\ell _{1})]^{T}}$

where ${\displaystyle \ell _{1}}$ is large enough and ${\displaystyle r_{W}(t)=Wr(t)}$, where ${\displaystyle W}$ is an appropriate filter.

The correlation function is:

${\displaystyle f_{N,\ell _{1}}(\rho )={\frac {1}{N}}\sum _{t=1}^{N}\zeta (t)\varepsilon (t,\rho )}$

and the optimization problem becomes:

${\displaystyle {\widehat {\rho }}={\underset {\rho \in D_{k}}{\operatorname {arg\,min} }}J_{N,\ell _{1}}(\rho )={\underset {\rho \in D_{k}}{\operatorname {arg\,min} }}f_{N,\ell _{1}}^{T}f_{N,\ell _{1}}.}$

Denoting with ${\displaystyle \phi _{r}(\omega )}$ the spectrum of ${\displaystyle r(t)}$, it can be demonstrated that, under some assumptions, if ${\displaystyle W}$ is selected as:

${\displaystyle W(e^{-j\omega })={\frac {F(e^{-j\omega })(1-M(e^{-j\omega }))}{\phi _{r}(\omega )}}}$

then, the following holds:

${\displaystyle \lim _{N,\ell _{1}\to \infty ,\ell _{1}/N\to \infty }{\widehat {\rho }}=\rho ^{*}.}$

Virtual Reference Feedback Tuning (VRFT) is a noniterative method for data-driven tuning of a fixed-structure controller. It provides a one-shot method to directly synthesize a controller based on a single dataset.

VRFT was first proposed in [4] and then extended to LPV systems.[5] VRFT also builds on ideas given in [6] as ${\displaystyle VRD^{2}}$.

The main idea is to define a desired closed loop model ${\displaystyle M}$ and to use its inverse dynamics to obtain a virtual reference ${\displaystyle r_{v}(t)}$ from the measured output signal ${\displaystyle y(t)}$.

The main idea is to define a desired closed loop model M and to use its inverse dynamics to obtain a virtual reference from the measured output signal y.

The virtual signals are ${\displaystyle r_{v}(t)=M^{-1}y(t)}$ and ${\displaystyle e_{v}(t)=r_{v}(t)-y(t).}$

The optimal controller is obtained from noiseless data by solving the following optimization problem:

${\displaystyle {\widehat {\rho }}_{\infty }={\underset {\rho }{\operatorname {arg\,min} }}\lim _{N\to \infty }J_{vr}(\rho )}$

where the optimization function is given as follows:

${\displaystyle J_{vr}^{N}(\rho )={\frac {1}{N}}\sum _{t=1}^{N}\left(u(t)-K(\rho )e_{v}(t)\right)^{2}.}$

• VRFT toolbox for MATLAB