Data-driven control systems are a broad family of control systems, in which the identification of the process model and/or the design of the controller are based entirely on experimental data collected from the plant.[1]
In many control applications, trying to write a mathematical model of the plant is considered a hard task, requiring efforts and time to the process and control engineers. This problem is overcome by data-driven methods, which fit a system model to the experimental data collected, choosing it in a specific models class. The control engineer can then exploit this model to design a proper controller for the system. However, it is still difficult to find a simple yet reliable model for a physical system, that includes only those dynamics of the system that are of interest for the control specifications. The direct data-driven methods allow to tune a controller, belonging to a given class, without the need of an identified model of the system. In this way, one can also simply weight process dynamics of interest inside the control cost function, and exclude those dynamics that are out of interest.
The standard approach to control systems design is organized in two-steps:
\widehat{G}=G\left(q;\widehat{\theta}N\right)
q
\widehat{\theta}N
G
N
\Gamma
G0
C
\widehat{G}
\widehat{G}
G0
\Gamma
One way to deal with uncertainty is to design a controller that has an acceptable performance with all models in
\Gamma
G0
In the following, the main classifications of data-driven control systems are presented.
There are many methods available to control the systems. The fundamental distinction is between indirect and direct controller design methods. The former group of techniques is still retaining the standard two-step approach, i.e. first a model is identified, then a controller is tuned based on such model. The main issue in doing so is that the controller is computed from the estimated model
\widehat{G}
\widehat{G} ≠ G0
Another important distinction is between iterative and noniterative (or one-shot) methods. In the former group, repeated iterations are needed to estimate the controller parameters, during which the optimization problem is performed based on the results of the previous iteration, and the estimation is expected to become more and more accurate at each iteration. This approach is also prone to on-line implementations (see below). In the latter group, the (optimal) controller parametrization is provided with a single optimization problem. This is particularly important for those systems in which iterations or repetitions of data collection experiments are limited or even not allowed (for example, due to economic aspects). In such cases, one should select a design technique capable of delivering a controller on a single data set. This approach is often implemented off-line (see below).
Since, on practical industrial applications, open-loop or closed-loop data are often available continuously, on-line data-driven techniques use those data to improve the quality of the identified model and/or the performance of the controller each time new information is collected on the plant. Instead, off-line approaches work on batch of data, which may be collected only once, or multiple times at a regular (but rather long) interval of time.
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
yd
r
\tilde{y}(\rho)=y(\rho)-yd
J(\rho)=
1 | |
2N |
N | |
\sum | |
t=1 |
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:
\rhoi+1=\rhoi-\gammai
-1 | |
R | |
i |
d\widehat{J | |
The value
\gammai
Ri
d\widehat{J | |
dJ | |
d\rho |
(\rho)=
1 | |
N |
N | ||
\sum | \left[\tilde{y}(t,\rho) | |
t=1 |
\deltay | |
\delta\rho |
(t,\rho)\right].
The value of
\deltay | |
\delta\rho |
(t,\rho)
C(\rho)
r
y(\rho)
y(1)(\rho)
C(\rho)
r
r-y(1)(\rho)
C(\rho)
y(2)(\rho)
\delta\widehat{y | |
A crucial factor for the convergence speed of the algorithm is the choice of
Ri
\tilde{y}
Ri=
1 | |
N |
N | |
\sum | |
t=1 |
\delta\widehat{y | |
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
G
M
F
K(\rho)=\betaT\rho
\rho\inRn
\beta
K*
M
G
The goal is to minimize the following objective function:
J(\rho)=\left\|F(
K*G-K(\rho)G | |
(1+K*G)2 |
)
2. | |
\right\| | |
2 |
J(\rho)
1 | |
(1+K(\rho)G) |
≈
1 | |
(1+K*G) |
When
G
\varepsilon(t)
The input signal
r(t)
v(t)
\varepsilon(t,\rho*)
r(t)
\rho
r(t)
\varepsilon(t,\rho*)
The vector of instrumental variables
\zeta(t)
\zeta(t)=[rW(t+\ell1),rW(t+\ell1-1),\ldots,rW(t),\ldots,rW(t-\ell1)]T
where
\ell1
rW(t)=Wr(t)
W
The correlation function is:
f | |
N,\ell1 |
(\rho)=
1 | |
N |
N | |
\sum | |
t=1 |
\zeta(t)\varepsilon(t,\rho)
and the optimization problem becomes:
\widehat{\rho}=\underset{\rho\inDk}{\operatorname{argmin}}
J | |
N,\ell1 |
(\rho)=\underset{\rho\inDk}{\operatorname{argmin}}
T | |
f | |
N,\ell1 |
f | |
N,\ell1 |
.
Denoting with
\phir(\omega)
r(t)
W
W(e-j\omega)=
F(e-j\omega)(1-M(e-j\omega)) | |
\phir(\omega) |
then, the following holds:
\lim | |
N,\ell1\toinfty,\ell1/N\toinfty |
\widehat{\rho}=\rho*.
There is no guarantee that the controller
K
J | |
N,\ell1 |
G
K*
K*
K(\rho)
K*=K(\rho)
\widehat{K}(\rho)
Consider a stabilizing controller
Ks
M | ||||
|
\Delta(\rho):=Ms-K(\rho)G(1-Ms)
\delta(\rho):=\left\|\Delta(\rho)\right\|infty.
Theorem
The controller
K(\rho)
G
\Delta(\rho)
\exist\deltaN\in(0,1)
\delta(\rho)\leq\deltaN.
Condition 1. is enforced when:
K(\rho)
K(\rho)
The model reference design with stability constraint becomes:
\rhos=\underset{\rho\inDk}{\operatorname{argmin}}J(\rho)
s.t.\delta(\rho)\leq\deltaN.
A convex data-driven estimation of
\delta(\rho)
Define the following:
\begin{align} &\widehat{R}r(\tau)=
1 | |
N |
N | |
\sum | |
t=1 |
r(t-\tau)r(t)for\tau=-\ell2,\ldots,\ell2\\[4pt] &\widehat{R}r\varepsilon(\tau)=
1 | |
N |
N | |
\sum | |
t=1 |
r(t-\tau)\varepsilon(t,\rho)for\tau=-\ell2,\ldots,\ell2. \end{align}
For stable minimum phase plants, the following convex data-driven optimization problem is given:
\begin{align} \widehat{\rho}&=\underset{\rho\inDk}{\operatorname{argmin}}
J | |
N,\ell1 |
(\rho)\\[3pt] &s.t.\\[3pt] &|
\ell2 | |
\sum | |
\tau=-\ell2 |
\widehat{R}r\varepsilon(\tau,\rho)
-j\tau\omegak | |
e |
|\leq\deltaN|
\ell2 | |
\sum | |
\tau=-\ell2 |
\widehat{R}r(\tau,\rho)
-j\tau\omegak | |
e |
|\\[4pt] \omegak&=
2\pik | |
2\ell2+1 |
, k=0,\ldots,\ell2+1. \end{align}
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
VRD2
The main idea is to define a desired closed loop model
M
rv(t)
y(t)
The virtual signals are
rv(t)=M-1y(t)
ev(t)=rv(t)-y(t).
The optimal controller is obtained from noiseless data by solving the following optimization problem:
\widehat{\rho}infty=\underset{\rho}{\operatorname{argmin}}\limNJvr(\rho)
where the optimization function is given as follows:
N | |
J | |
vr |
(\rho)=
1 | |
N |
N | |
\sum | |
t=1 |
\left(u(t)-K(\rho)ev(t)\right)2.
An Introduction to Data-Driven Control SystemsAli Khaki-Sedigh
ISBN: 978-1-394-19642-5 November 2023 Wiley-IEEE Press 384 Pages
H2