## Abstract

Fractional-order differentiation offers new degrees of freedom that simplify the design of high-performance dynamic controllers. The CRONE control system design (CSD) methodology proposes the design of robust controllers by using fractional-order operators. A software toolbox has been developed based on this methodology and is freely available for the international scientific and industrial communities. This paper presents both the CRONE CSD methodology and its implementation using the toolbox. The design of two robust controllers for irrigation canals shows how to use the toolbox.

## 1. Introduction

The CRONE (Commande Robuste d’Ordre Non Entier) toolbox, developed gradually since the 1990s [1] by the CRONE research group, is a Matlab and Simulink toolbox dedicated to applications of fractional (or non-integer) derivatives in engineering and science. The toolbox is intended not only for researchers, including PhD students, but also for the industrial community, whose growing interest in fractional systems and whose readiness to invest in the development of applications are manifest. Original theoretical mathematical concepts developed in the group were integrated in the first version of the CRONE toolbox [1] as four main modules, each of which deals with a specific application field of fractional derivatives: the mathematical module contains several algorithms of fractional calculus; the system-identification module contains frequency- and time-domain system-identification algorithms; the CRONE control system design module uses fractional derivatives as high-level design parameters to simplify the design of robust control systems; and the path-planning module contains algorithms to optimize paths in dynamic environments. The CRONE toolbox is currently evolving towards an object-oriented version, which allows many enhancements [2]. Three main user classes, dedicated to fractional system representations, namely fractional transfer functions (`frac_tf`), fractional zeros, poles and gain (`frac_zpk`) and fractional state space (`frac_ss`), are developed. All three user classes are children of a parent class (`frac_lti`), which contains some common attributes of fractional systems. Among enhancements of the object programming of the CRONE toolbox is the overloading of basic operators (+,−,×,/,…) and standard Matlab scripts (`lsim`, `bode`, `nichols`, …) for the newly developed classes. As a consequence, an end-user familiar with standard Matlab operators and scripts can use the CRONE toolbox straightforwardly.

During the last 10 years, the CRONE toolbox has only been available for some privileged researchers and engineers, who granted its development (CNRS and PSA car company). On the other hand, the last two decades have witnessed a growing interest in fractional derivatives and their applications [3,4,5,6,7]. From now on, the CRONE toolbox is freely available for the international scientific community for research and pedagogical purposes. The aim of the CRONE group is to share its developments and its knowledge with scientists, researchers and engineers worldwide. The CRONE toolbox can be downloaded at http://cronetoolbox.ims-bordeaux.fr.

This paper presents the CRONE control system design (CSD) methodology, and how several tools of the CRONE CSD toolbox can be used to design single-input single-output (SISO) robust controllers for time-delay perturbed plants. Irrigation canals are chosen as an example for the CRONE CSD because their models are perturbed and include large time delays.

## 2. CRONE control system design methodology

The CRONE CSD methodology is a frequency-domain approach developed since the 1980s [8,9,10,11,12]. It is based on the common unity-feedback configuration. Three CRONE CSD methods have been developed, successively extending the application field. In these three methods, the controller or open-loop transfer function is defined using integro-differentiation of fractional order.

### (a) First-generation CRONE control system design method

The first-generation CRONE CSD method is based on a constant-phase controller (around the required open-loop gain crossover frequency *ω*_{cg}) obtained through a band-limited real fractional differentiator
2.1
When the plant phase is constant around the frequency *ω*_{cg}, in the presence of plant perturbations that induce gain variations only (around *ω*_{cg}), the constant-phase controller ensures the robustness of the phase margin.

For the CRONE controller to handle the control effort level and the steady-state error, it has to include an order-*n*_{I} band-limited integrator and an order-*n*_{F} low-pass filter. Consequently, the first-generation CRONE controller is defined as
2.2
with , and . It can be considered as a PI^{nI}D^{n}F^{nF} controller whose implementable version, *C*_{R}(*s*), defined using a rational approximation of the fractional part of (2.2), is obtained using a recursive distribution of real-valued zeros and poles:
2.3
and
2.4
The perfect robustness of the phase margin is ensured only if *ω*_{cg} is within a frequency range where the plant phase is constant (only gain-like plant variation may occur). When this condition is not met, the second-generation CRONE CSD method is used.

### (b) Second-generation CRONE control system design method

For control effort-level problems, it is sometimes impossible to choose an *ω*_{cg} within a frequency range where the plant phase is constant. Nevertheless, as Bode [13] first stated for the ‘design of single loop absolutely stable amplifiers’, whose tube gains vary, the robust controller is the one which permits the open-loop transfer function to be defined by a constant phase in a useful band. So, the second-generation CRONE approach defines the open-loop transfer function around *ω*_{cg} as the transfer function of a fractional-order integrator
2.5
The vertical sliding of the vertical Nichols locus (called vertical template), depending on gain variations, allows one to have a constant (i) phase margin, (ii) resonant peak *M*_{r}, (iii) modulus margin (the minimal distance between the open-loop Nyquist locus and the critical point), and (iv) damping ratio of the closed-loop system. For the controller to handle accuracy, robustness and control effort problems, using integer orders *n*_{l} and *n*_{h}, the nominal open-loop transfer function is defined by
2.6
with *n*_{l} and . Order *n*_{l} has to be set to handle the accuracy provided by the control system. Order *n*_{h} has to be set to obtain a proper or bi-proper controller.

The fractional controller *C*_{F}(*s*) is defined as the ratio of the nominal open-loop and the plant transfer functions
2.7
Finally, the rational (i.e. with integer orders) transfer function *C*_{R}(*s*) can be defined by replacing the fractional part of (2.7) using the rational approximation (2.3).

### (c) Third-generation CRONE control system design method

For cases more general than gain-like plant variations, the vertical Nichols locus does not slide on itself. It is more convenient to consider a Nichols locus that is still defined as a straight-line segment (around the frequency *ω*_{cg}) for the nominal parametric state of the plant, but with any direction. The open-loop transfer function is now defined as the real part with respect to the imaginary unit i of a complex fractional-order integrator transfer function:
2.8
with *n*=*a*+i*b*∈*C*_{i} and *s*=*σ*+j*ω*∈*C*_{j}. The real order *a* determines the phase placement of the Nichols locus (called generalized template), and then the imaginary order *b* determines its angle with respect to the vertical. Among the infinite number of these templates, which tangent the magnitude contour related to a required magnitude peak *M*_{r0} for the nominal parametric state of the plant, the *optimal template* can be defined as the one which permits minimizing of the variations of *M*_{r} stemming from the frequency uncertainty domains which in turn stem from the various frequency responses of the perturbed plant. The principle of the third-generation CRONE control methodology is to optimize the parameters of the nominal open-loop transfer function *β*_{0}(*s*) that includes a band-limited complex fractional-order integration
2.9
where *β*_{m}(*s*) is a set of band-limited generalized templates:
2.10
with
2.11
where
2.12
where *N*^{+},*N*^{−} and ; and *ω*_{r},*ω*_{k},*ω*_{k+1},*α*_{k},*C*_{k},*C*_{l},*C*_{h},*a*_{k} and .

Gains *C*_{x} are chosen such that *ω*_{r} is the nominal closed-loop resonant frequency. When it is useful, *N*^{−} and *N*^{+} are different from 0 to increase the number of tuning parameters.

As the open-loop parameters are optimized in order to reduce the variation of the resonant peak *M*_{r} of the complementary sensitivity function *T*(*s*), the robustness cost function to be minimized could be
2.13
where *M*_{r0} is the required value of the nominal closed-loop resonant peak (for the nominal plant *G*_{0}), while respecting the following set of inequality constraints for all plants *G* and for :
2.14
2.15
It is easy to show that the multiplicative uncertainty *Δ*_{m}*β*(*s*) of the open-loop frequency response, which defines the frequency-domain uncertainty in the Nichols chart, is invariant and equals that of the plant
2.16
where *G*_{0}(*s*) and *β*_{0}(*s*) are the nominal plant and open-loop transfer functions and where *Δ*_{m}*G*(*s*) (or *Δ*_{m}*β*(*s*)) is a multiplicative uncertainty model. Thus, the uncertainty frequency domains related to the Nichols locus of *β*_{0}(j*ω*) are defined by all the possible values of the ordered pair . By minimizing *J* (2.13), the optimal open-loop parameters position the uncertainty frequency domains so that they overlap as little as possible with the low-stability margin areas of the Nichols chart. Since the uncertainties are taken into account by the least conservative method, only a nonlinear optimization method can be used. For *N*^{−}=*N*^{+}=0, only four independent open-loop parameters have to be optimized. The parametrization of the open-loop transfer function by complex fractional orders then simplifies the optimization considerably. During optimization, a complex order plays the role of a large set of parameters found in common rational controllers. Finally, the fractional controller *C*_{F}(*s*) is defined by its frequency response
2.17
The parameters of a rational transfer function *C*_{R}(*s*) with a predefined low-order structure are tuned to fit the ideal frequency response *C*_{F}(j*ω*) using any frequency-domain system-identification technique. An advantage of this design method is that, whatever the complexity of the control problem, a reduced value of the controller order (usually around 6) can be used.

### (d) CRONE control system design for particular plants

The rational form of the controller is obtained from its ideal frequency response defined by (2.17) and the constraints it needs to satisfy, such as:

— the internal closed-loop stability when the plant is nominal;

— the closed-loop stability when the plant is perturbed;

— the achievability of the rational controller; and

— a good damping of signals for a low-damped plant.

Thus, before any optimization, the nominal open-loop transfer function *β*_{0}(*s*) needs to be modified by including the nominal right half-plane zeros and poles, the time delay and the under-damped modes of the nominal plant
2.18
where z_{i} is one of the *n*_{z} right half-plane zeros, *τ* the time-delay, p_{i} one of the *n*_{p} right half-plane poles, the natural frequency and the damping coefficient (*ω*_{nzi},*ς*_{zi}) define one of the *n*_{z} numerator under-damped modes; the natural frequency and the damping coefficient (*ω*_{npi},*ς*_{pi}) define one of its *n*_{p} denominator under-damped modes.

### (e) CRONE control system design toolbox

The CRONE CSD toolbox for Matlab has been developed to make it as easy as possible to design robust SISO controllers. The perturbed plant can be defined in several ways:

— using a formal linear time-invariant (LTI) model with uncertain parameters;

— using a set of measured/computed frequency responses;

— using a set of transfer functions; and

— using a Simulink SISO or multi-input multi-output (MIMO) model with uncertain parameters.

Then a guided start tool permits:

— the definition of the controller implementation way (continuous time or discrete time);

— the definition of very simple specifications (step response settling time and overshoot, integral and low-pass controller effect);

— the analysis of specifications and plant (required bandwidth, type of plant perturbation, particularities of the nominal plant) in order to propose the choice of one of the three CRONE CSD methods and tools; and

— the pre-parametrization of the chosen CRONE CSD tool.

Each of the three CRONE CSD tools proposes:

— to plot the frequency responses of the plant, the controller, the open-loop and the sensitivity functions;

— to tune the fractional-order controller or the open-loop transfer function parameters;

— to synthesize the rational controller and to build its Simulink model;

— to compute the frequency-domain performance; and

— to simulate the time-domain closed-loop responses.

This paper proposes to present the way the toolbox can be used to design robust controllers for two irrigation canals.

## 3. Robust control of two irrigation canals

Time-delay systems are frequently used to present the design of proportional integral derivative (PID) controllers and particularly of fractional-order PID controllers. Feliu-Batlle *et al.* [14] proposed the design of robust fractional-order PID controllers for two irrigation canals. The first system models a 10 km-long section of the 60 km-long Güira de Melena Main Irrigation Canal (GMMIC) in Habana County, Cuba. The controller has to control the downstream end water level of one section by using its upstream gate position. An order-2 transfer function that includes a time delay is proposed:
3.1
The parameters of this linear model vary with the water flow rate for various upstream and downstream water levels. The uncertain parameters are defined by
3.2
The upstream gate dynamic is invariant with *T*=60 s.

The second system models a 5 m-long prototype canal of the Fluids Mechanics Laboratory of the Castilla-La Mancha University, Spain [14,15]. It is defined by a first-order transfer function and a time delay: 3.3 The uncertain parameters are defined by 3.4

### (a) Control system design for the GMMIC canal

A Matlab `plant_cubanCC3.mat` file that defines the perturbed plant can be created by using the `ltimodel2plantfile` function of the CRONE CSD toolbox:

`ltimodel2plantfile(’plant_cuban’,’p(1)/p(2)*s+1)/(60*s+1)`

`*exp(-p(3)*s)’,[1.875,6000,1800],...`

`[1.25,6,0],[3.75,6000,1800],logspace(-6,0,100)`,

`1,[5,5,5],[’log’,’log’,’lin’])`;

The second input argument is the LTI definition of the perturbed plant. The other arguments define: the nominal parameter values, the extreme parameter values, the frequency range chosen for the design, a figure number to plot the Nichols locus of the perturbed plant (figure 1) and the number and distribution type used to grid the parameter variations. The created file contains the plant nominal frequency response, a plant frequency uncertainty domain for each frequency, the plant nominal time delay, and the right half-plane or under-damped plant nominal zeros and poles if any.

Using the guided start tool (figure 2), the controller is chosen in continuous time and the settling time about 10 000 s with a 20 per cent overshoot for a step response (with respect to the reference input). The analysis function explains that: the nominal open-loop gain crossover frequency is around 0.0003 rad s^{−1}; at gain crossover frequency, the uncertainty of the plant frequency response is 15.93 dB for the magnitude and 92.76^{°} for the phase; the open-loop gain crossover frequency can vary from 0.000 22 to 0.000 83 rad s^{−1} and within this frequency range the variation of the plant phase equals at least 100.4^{°}.

Thus, from the given specifications and the analysis of the nominal plant, the third-generation CRONE CSD tool (figure 3) proposed for the design of the robust controller is pre-parametrized. For instance, the open-loop parameters are initialized to *n*_{l}=1, *n*_{h}=3 and *N*^{+}=*N*^{−}=0 for constant parameters; *ω*_{r}=0.0003 rad s^{−1}, *Y* _{0}=|*β*_{0}(j*ω*_{r})|=2 dB, *ω*_{0}=*ω*_{r}/5 and *ω*_{1}=5*ω*_{r} for the four parameters to be optimized.

The constraints on the four common sensitivity functions (2.14) are initialized as in figure 4. Figures 4 and 5 show that the initial parameters of the nominal open-loop transfer function have to be modified in order to reduce resonant peaks.

The constraints can be modified in order to improve the designed controller. To manage the time-delay effect, parameter *N*^{+} of (2.10) is changed to 1. The three additional parameters to be optimized are initialized to *ω*_{2}=2*ω*_{1}, *a*_{2}=2 and *b*_{2}=0.

The seven (4+3) open-loop parameters can be modified by the user (figure 6) or automatically optimized (figure 7). Figures 7 and 8 show that the sensitivity functions almost respect the constraints. The least case closed-loop resonant peak *M*_{r} only exceeds the nominal required value (*M*_{r0}=1.74 dB) of 1.27 dB.

The rational transfer function *C*_{R}(*s*) of the controller is obtained through:

— the automatic initialization of its gain, number of differentiator, and number and value of its zeros and poles;

— the aided tuning of its parameters (figure 9); and

— the automatic optimization of the parameters.

Figure 9 shows that the frequency response of *C*_{R}(*s*) correctly fits the desired one for an order-5 transfer function.

Figure 10 shows the continuous-time (as initially required) Simulink model of the controller that is automatically built.

This controller is then used to simulate the time-domain response of irrigation canal models (figure 11).

For the set of models defined by (3.1) and (3.2), the least case performance is defined by

— settling time

*t*_{5%}≤4.36×10^{4}s;— overshoot ≤ 31%; and

— ISE (integral squared error) ≤ 8566.

These values cannot really be compared with those presented by Feliu-Batlle *et al.* [14], as their results were only given for *τ*_{1}=600 s.

### (b) Control system design for prototype canal

The perturbed plant is now defined with:

`ltimodel2plantfile(’plant_spain’,’p(1)/(p(2)*s+1)*exp(-p(3)*s)’`,

`[0.2,25.33,4.3],[0.14,...9.62,3.1]`,

`[0.26,25.33,4.3],logspace(-3,2,100)`,

`1,[5,5,5],[’log’,’log’,’lin’])`

As the plant is a first-order system, the high-frequency open-loop order is now set to *n*_{h}=2. As the time delay is smaller, the bandwidth could be greater. Figure 12 shows the frequency response and parameters of the optimal open-loop fractional-order transfer function. The closed-loop resonant peak *M*_{r} exceeds the nominal required value of 1.05 dB.

An order-6 transfer function is used to synthesis *C*_{R}(*s*). It is used to simulate the time-domain response (figure 13) of irrigation canal for the set of models defined by (3.3) and (3.4). The least case performance is defined by:

— settling time

*t*_{5%}≤52.84 s;— overshoot ≤27%; and

— ISE≤9.97.

Using several fractional-order PID controllers, improved with integer-order integrators, and minimizing the aggregating ISE cost function, Calderon-Valdez *et al.* [15] obtained a settling time *t*_{5%}<64 s. It proves that the proposed CRONE CSD toolbox can provide good performance. Furthermore, it is simple to use and allows us to design controllers for a very large set of perturbed plants.

## Footnotes

One contribution of 14 to a Theme Issue ‘Fractional calculus and its applications’.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved