\[ \def\A{\mathrm{{\bf A}}} \def\B{\mathrm{{\bf B}}} \def\C{\mathrm{{\bf C}}} \def\D{\mathrm{{\bf D}}} \def\x{\mathrm{{\bf x}}} \def\y{\mathrm{{\bf y}}} \def\u{\mathrm{{\bf u}}} \def\dif{\mathrm{d}} \def\euler{\mathrm{e}} \def\ivlt#1{[\underline{#1},\overline{#1}]} \def\ivl#1#2{[#1,#2]} \]

IEEE Estonia Section Annual Seminar | Viinistu Art Hotel | Sep 6, 2024

Some Recent Advances in
Robust Fractional-order Control

Aleksei Tepljakov$^{\star}$ & Majid Ghorbani

Centre for Intelligent Systems, https://cis.taltech.ee/
Department of Computer Systems, School of Information Technologies,
Tallinn University of Technology, Estonia

About the Speaker: Aleksei Tepljakov

And briefly about background of CIS...

Talk outline

  • Brief introduction to fractional calculus, fractional models and FOPID controllers.
  • Recent advances in robust fractional control with scientific software support.
  • Conclusions and further perspectives.

Origin of Fractional Calculus

Leibniz

Hey Guillaume, quick question. Can the meaning of derivatives with integer orders $D^n / D x^n $ be generalized to those with non-integer orders?

L'Hopital

Hey, interesting thought. What if $n = \frac{1}{2} $?

Leibniz

🤔 That would lead to a paradox! But one day useful consequences might be drawn from it.

Then...

Fractional calculus is a generalization of integration and differentiation to the non-integer order operator $_{a}\mathscr{D}{}_{t}^{\alpha}$, where $a$ and $t$ are the lower and upper bounds of the operation, $\alpha\in\mathbb{R}$ denotes the fractional order such that

\begin{equation} _{a}\mathscr{D}_{t}^{\alpha}=\begin{cases} \frac{\dif^{\alpha}}{\dif t^{\alpha}} & \alpha>0,\\ 1 & \alpha=0,\\ \int_{a}^{t}\left(\dif\tau\right)^{-\alpha} & \alpha<0. \end{cases} \end{equation}

If $\alpha$ is an integer number, the definition corresponds to a classical differentiation and integration operation.

Fractional-order Differential Equations

A linear, fractional-order continuous-time dynamic system can be expressed by a fractional differential equation of the following form

\begin{multline} a_{n}\mathscr{D}^{\alpha_{n}}y(t)+a_{n-1}\mathscr{D}^{\alpha_{n-1}}y(t)+\cdots+a_{0}\mathscr{D}^{\alpha_{0}}y(t) = \\b_{m}\mathscr{D}^{\beta_{m}}u(t)+b_{m-1}\mathscr{D}^{\beta_{m-1}}u(t)+\cdots+b_{0}\mathscr{D}^{\beta_{0}}u(t), \end{multline}

where $a_{k},\,b_{k}\in\mathbb{R}$. The system is said to be of commensurate-order if in the above all the orders of derivation are integer multiples of a base order $\gamma$ such that $\alpha_{k},\,\beta_{k}=k\gamma,\,\gamma\in\mathbb{R}_{+}$. The system can then be expressed as

\begin{equation} \sum_{k=0}^{n}a_{k}\mathscr{D}^{k\gamma}y(t)=\sum_{k=0}^{m}b_{k}\mathscr{D}^{k\gamma}u(t).\label{eq:TfComOrd} \end{equation}

Fractional-order Transfer Functions

Applying the Laplace transform to the presented FODE with zero initial conditions the input-output representation of the fractional-order system can be obtained in the form of a transfer function:

\begin{equation} G(s)=\frac{Y(s)}{U(s)}=\frac{b_{m}s^{\beta_{m}}+b_{m-1}s^{\beta_{m-1}}+\cdots+b_{0}s^{\beta_{0}}}{a_{n}s^{\alpha_{n}}+a_{n-1}s^{\alpha_{n-1}}+\cdots+a_{0}s^{\alpha_{0}}}.\label{eq:Gs} \end{equation}

In the case of a system with commensurate order $\gamma$ we have

\begin{equation} G(s)=\frac{\sum\limits _{k=0}^{m}b_{k}\left(s^{\gamma}\right)^{k}}{\sum\limits _{k=0}^{n}a_{k}\left(s^{\gamma}\right)^{k}}.\label{eq:GsComOrd} \end{equation}

Approximation of fractional operators

The Oustaloup recursive filter gives a very good approximation of fractional operators in a specified frequency range and is widely used in applied fractional calculus. For a frequency range $(\omega_{b},\,\omega_{h})$ and of order $N$ the filter for an operator $s^{\gamma},\,0<\gamma<1$, is given by

\begin{eqnarray} s^{\gamma} & \approx & K\prod_{k=-N}^{N}{\frac{s+\omega_{k}^{\prime}}{s+\omega_{k}}},\quad K=\omega_{h}^{\gamma},\quad\omega_{r}=\frac{\omega_{h}}{\omega_{b}},\label{eq:OustaFod}\\ \omega_{k}^{\prime} & = & \omega_{b}(\omega_{r})^{\frac{k+N+\frac{1}{2}(1-\gamma)}{2N+1}},\quad\omega_{k}=\omega_{b}(\omega_{r})^{\frac{k+N+\frac{1}{2}(1+\gamma)}{2N+1}}.\notag \end{eqnarray}

The resulting integer-order model order is $2N+1$.

Approximation example

The fractional-order transfer function is

\begin{equation} G(s)=\frac{1}{14994s^{1.31}+6009.5s^{0.97}+1.69}, \end{equation}

and approximation parameters $\omega=[10^{-4}; 10^4]$, $N=5$. This system is approximated using conventional and modified Oustaloup filter variants. Results are provided graphically below.

Fractional-order PID controllers

The control law of the PI$^{\lambda}$D$^{\mu}$ controller can be expressed as follows:

\begin{equation} u(t)=K_{p}e(t)+K_{i}\mathscr{D}^{-\lambda}e(t)+K_{d}\mathscr{D}^{\mu}e(t),\label{eq:PIDCtrlAct} \end{equation}

where $e(t)=y_{sp}(t)-y(t)$ is the error signal. After applying the Laplace transform to the above assuming zero initial conditions, the following equation is obtained:

\begin{equation} C(s)=K_{p}+\frac{K_{i}}{s^{\lambda}}+K_{d}s^{\mu}\label{eq:GCPid} \end{equation}

Obviously, when taking $\lambda=\mu=1$ the result is the classical integer-order PID controller.

Fractional control actions

Fractional integrating control action

Fractional differentiating control action

Industrial applicability of
FOPID controllers

We suggest to review the following paper

Additional theoretical settings

Interval Arithmetic and
Interval Uncertainty

Simple examples

Addition:

\[ [1, 2] + [3, 4] = [1 + 3, 2 + 4] = [4, 6]. \]

Multiplication:

\[ [1, 2] \cdot [3, 4] = [\min(S), \max(S)] = [3, 8], \]

where $S = \{ 1\cdot 3, 1\cdot 4, 2\cdot 3, 2\cdot 4 \} = \{3, 4, 6, 8\}$.

Interval uncertainty in polynomials

Using the idea of intervals, we can introduce the concept of interval uncertainties. We can use the interval uncertainty structure to define polynomials with uncertain coefficients falling into known intervals:

\[ p(x) = \ivlt{a_n}x^n + \ivlt{a_{n-1}}x^{n-1} + \dots + \ivlt{a_1}x + \ivlt{a_0}. \]

where $\ivlt{a}$ denote intervals with a lower bound $\underline{a}$ and an upper bound $\overline{a}$.

Obviously, we can also introduce this concept to transfer functions, and further, to fractional-order transfer functions.

Robust Stability

Thus, we can define a fractional-order transfer function with interval uncertaintites:

\[ G(s) = \frac{ \ivlt{b_m} s^{\beta_m} + \ivlt{b_{m-1}} s^{\beta_{m-1}} + \dots + \ivlt{b_1} s^{\beta_1} + \ivlt{b_0} s^{\beta_0} } { \ivlt{a_n} s^{\alpha_n} + \ivlt{a_{n-1}} s^{\alpha_{n-1}} + \dots + \ivlt{a_1} s^{\alpha_1} + \ivlt{a_0} s^{\alpha_0} }, \]

where for process models it is usually assumed that $\beta_0 = \alpha_0 = 0$.

Just as with regular systems, we study the characteristic polynomial of the system or of the system composition with a controller. We say that the system is robustly stable, if we can prove that the corresponding system having interval uncertainties is stable.

The contribution

FOMCON Toolbox and
New Developments

Structure of the toolbox (2024)

New classes in FOMCON

  • ufpoly — (fractional) polynomial with coefficients and/or exponents that can be represented with uncertainty using intervals.
  • ufotf — (fractional) transfer function having polynomials with uncertain coefficients/exponents represented by intervals, and possibly an uncertain delay term.

Simple example

Input the following fractional-order transfer function:

\[ \begin{equation} G_{1u} = \frac{s^{[0.1, 0.2]}+1}{[1, 2]s^{[1.5, 1.7]}+3s^{[0.5, 0.55]}-10}. \end{equation} \]

In MATLAB, use the following code to get:

>> G1u = ufotf('s^[0.1,0.2] + 1', ...
            '[1,2]s^[1.5,1.7] + 3s^[0.5,0.55] - 10')

            s^{[0.1, 0.2]}+1
----------------------------------------
[1, 2]s^{[1.5, 1.7]}+3s^{[0.5, 0.55]}-10

Fractional-order transfer function with uncertainty intervals.

Worked example

Problem Setting

Consider the following interval plant

\[ \begin{aligned} &P(s)=\frac{[1, 2] s + [1, 2]}{s^2 + [1, 1.5] s + [0.5, 1.5]} \end{aligned} \]

and a FOPID controller

\[ \begin{aligned} C(s) = 2 + \frac{1.8}{s} + 2.6 s^{0.1}. \end{aligned} \]

Our objective is to determine whether the FOPID controller can robustly stabilize the negative unity feedback control system in the presence of interval uncertainties.

Let's find the characteristic polynomial

This is done in a conventional way, but using interval arithmetic. Finally, we arrive at the following expression:

\[ \begin{multline} \Delta(s) = s^3 + [2.6,5.2] s^{2.1} +[3, 5.5] s^2 \\ + [2.6,5.2] s^{1.1} + [4.3,9.1] s+[1.8,3.6]. \end{multline} \]

We will now perform further steps on this representation using MATLAB code from the new feature set.

Creating a polynomial and sampling it

First, we create the ufpoly object, then sample it, and obtain a Mikhailov plot:

>> F = ufpoly(['s^3 + [2.6, 5.2]s^2.1 + [3, 5.5]s^2 + [2.6, 5.2]s^1.1' ...
            '+ [4.3, 9.1]s + [1.8, 3.6]'])

s^{3}+[2.6, 5.2]s^{2.1}+[3, 5.5]s^{2}+[2.6, 5.2]s^{1.1}+[4.3, 9.1]s+[1.8, 3.6]

Fractional polynomial with uncertainty intervals.

>> Fs = sample(F)  % Result will be different every time

s^{3}+3.862s^{2.1}+5.0007s^{2}+2.9689s^{1.1}+6.3245s+3.4483

Fractional polynomial.

>> mikhailovfo(Fs)

The resulting Mikhailov plot

Further analysis

The sampled member $F_s$ is stable, as the origin is not encircled. For that reason, we continue testing robust stability of the system with interval uncertainties.

We use the following commands in MATLAB (two different approaches are considered):

>> robstabfo1(F)
    "robustly stable"

>> F1 = ufpoly('s^3+[2.6, 5.2]s^2.1+[3, 5.5]s^2');
>> F2 = ufpoly('[2.6, 5.2]s^1.1+[4.3, 9.1]s+[1.8, 3.6]');
>> robstabfo2(F1, F2)
    "robustly stable"

According to the above, the system described previously is found to be robustly stable.

Further elaboration

The technical details are presented in the mentioned paper. Please refer to it in case of interest.

 

Conclusions and future outlooks

  • Interval arithmetic and the interval uncertainty structure provide an excellent foundation for engineering robust control systems.
  • FOMCON toolbox for MATLAB is the most popular FO modeling and control toolbox with hundreds of monthly downloads from MathWorks File Exchange and thus is an excellent fit to host these new developments.
  • The particular topic is of high interest to the FO community and many requests have been made already to include relevant results in FOMCON.
  • Future work thus includes the complete implementation of mentioned classes and support in FOMCON and publishing relevant research papers.

Thank you for your attention!