Fractional Order Motion Controls

By Ying Luo and Yang Chen

Covering fractional order theory, simulation and experiments, this book explains how fractional order modelling and fractional order controller design compares favourably with traditional velocity and position control systems. The authors systematically compare the two approaches using applied fractional calculus. Stability theory in fractional order controllers design is also analysed.

•  Presents material suitable for a variety of real-world applications, including hard disk drives, vehicular controls, robot control and micropositioners in DNA microarray analysis
• Includes extensive experimental results from both lab bench level tests and industrial level, mass-production-ready implementations
• Covers detailed derivations and numerical simulations for each case
• Discusses feasible design specifications, ideal for practicing engineers

The book also covers key topics including: fractional order disturbance cancellation and adaptive learning control studies for external disturbances; optimization approaches for nonlinear system control and design schemes with backlash and friction. Illustrations and experimental validations are included for each of the proposed control schemes to enable readers to develop a clear understanding of the approaches covered, and move on to apply them in real-world scenarios.

• Language: English
• Publisher: Wiley
• Release date: Nov 7, 2012
• ISBN: 9781118387719

Ying Luo

Dr. Luo currently working in National Lab of Radar Signal Processing, Xidian University as a Postdoctoral Fellow, and also is an Adjunct Research Fellow with the Key Laboratory for Information Science of Electromagnetic Waves (Ministry of Education), Fudan University, Shanghai, China. He is a Member of the IEEE and the Chinese Institute of Electronics (CIE), respectively. He has published over 60 papers on journals and conferences. His research interests include signal processing and auto target recognition (ATR) in SAR and ISAR.

Book preview

Part I

Fundamentals of Fractional Order Controls

1

Introduction

It is known that the nth order derivative of a function f(t) can be mathematically described by dny/dxn. With this notation, one may ask "What does n = 1/2 mean in the notation? Actually, this was the question asked in a letter by the French mathematician Guillaume François Antoine L’Hôpital to one of the inventors of calculus, the French mathematician Gottfried Wilhelm Leibnitz more than 300 years ago. In answering to the letter, Leibnitz said: It will lead to a paradox, from which one day useful consequences will be drawn." This marks the beginning of fractional calculus. However, earlier research concentrated on theoretical math issues. Fractional calculus is now being widely used in many areas. For instance, in the discipline of automatic control, fractional order control is a promising new topic [161].

1.1 Fractional Calculus

The idea of Fractional Calculus has been known since the development of the regular (integer order) calculus, with the first reference probably being associated with Leibniz and L’Hôpital in 1695 where the half-order derivative was mentioned.

Fractional calculus is a generalization of integration and differentiation to the non-integer order fundamental operator aDrt, where a and t are the limits of the operation. The continuous integro-differential operator is defined as

Unnumbered Display Equation

where r is the order of the operation, generally inline but r could also be a complex number [179].

1.1.1 Definitions and Properties

Various definitions have appeared in the development and studies of fractional calculus. Some of the definitions are directly extended from the conventional integer order calculus. The commonly used definitions are summarized as follows [161]:

A. Fractional order Cauchy integral formula

The formula is extended from integer order calculus

(1.1) Numbered Display Equation

where C is the closed-path that encircles the poles of the function f(t).

The integrals and derivatives for sinusoidal and cosine functions can be expressed by

(1.2)

Numbered Display Equation

It can also be shown with Cauchy’s formula that, if k is not an integer, the above formula is still valid.

B. Grünwald-Letnikov definition

The fractional order differentiation and integral can be defined in a unified way such that

(1.3)

Numbered Display Equation

where inline are the binomial coefficients; the subscripts to the left and right of D are the lower- and upper-bounds in the integral. The value of α can be positive or negative, corresponding to differentiation and integration, respectively and the α is non-integer.

C. Riemann-Liouville definition

The fractional order integral is defined as

(1.4) Numbered Display Equation

where 0<α <1, and a is the initial value. Let a = 0, the notation of integral can be simplified to D−αtf(t). The Riemann-Liouville definition is a widely used definition for fractional order differentiation and integral. Similarly, fractional order differentiation is defined as

(1.5)

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where inline .

D. Caputo definition

The Caputo fractional order differentiation is defined by

(1.6) Numbered Display Equation

where α = m+γ, m is an integer and inline . Similarly, by the Caputo definition, the integral is described by

(1.7) Numbered Display Equation

It can be shown that for a great varieties of functions, the Grünwald-Letnikov and the Riemann-Liouville definitions are equivalent [188].

The properties of fractional calculus are summarized as below [188]:

A. If f(t) is an analytical function of t, its fractional derivative 0Dαtf(t) is an analytical function of t and α.

B. For α = n, where n is an integer, the operation 0Dαtf(t) gives the same result as classical differentiation of integer order n.

C. For α = 0, the operation 0Dαtf(t) is the identity operator:

unnumbered Display Equation

D. Fractional differentiation and fractional integration are linear operations:

unnumbered Display Equation

E. The additive index law (semigroup property)

unnumbered Display Equation

holds under some reasonable constraints on the function f(t).

The fractional order derivative commutes with integer order derivative

unnumbered Display Equation

under the condition t = a, we have inline . The relationship above says the operators inline and aDrt commute (see [189, Chapter 2] for other commute properties).

1.1.2 Laplace Transform

Consider the linear fractional order differential equation given by

(1.8)

Numbered Display Equation

If all the initial values of the input and output are zero, the Laplace transform can be applied such that the differential equation can be mapped into an algebraic equation, from which the fractional order transfer function can be defined

(1.9)

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The Fourier transform for the fractional order derivative and integral can be defined in a unified way as

(1.10) Numbered Display Equation

where α can be either a positive or negative real number. The Laplace transform of a fractional order integral can be expressed by

(1.11) Numbered Display Equation

and the transform for a fractional order derivative (Riemann-Liouville definition) can be evaluated from

(1.12)

Numbered Display Equation

where, inline . In particular, if the derivatives of the function f(t) at t = a are all equal to 0, one simply has inline .

1.1.3 Fractional Order Dynamic Systems

Many real dynamic systems are better characterized using a non-integer order dynamic model based on fractional calculus or differentiation or integration of non-integer order. Traditional calculus is based on integer order differentiation and integration. The concept of fractional calculus has tremendous potential to change the way we see, model, and control the world around us.

Fractional calculus is a topic more than 300 years old. The number of applications where fractional calculus has been used is rapidly growing. These mathematical phenomena describe a real object more accurately than the classical integer order methods. The real objects are generally fractional [163], [178], [189], 234], however, for many of them the fractionality is very low. A typical example of a non-integer (fractional) order system is the voltage-current relation of a semi-infinite lossy transmission line [228] or the diffusion of the heat into a semi-infinite solid, where, under coefficients normalized to unity, the heat flow q(t) in nature is equal to the semi-derivative of the temperature T(t) [188], [190]

unnumbered Display Equation

Clearly, using an integer order ordinary differential equation (ODE) description for the above system may differ significantly to the actual situation. However, the fact that the integer order dynamic models are more welcome is probably due to the absence of solutions for fractional order differential equations (FODEs). At present, there are lots of methods for the approximation of fractional derivative and integral equations, therefore, some progress in the analysis of dynamic systems modeled by FODEs has been made in [3], [15], [19], [69], [152], [169], [2], [187], [188], [233], [246]. Recently, fractional calculus can be easily used in wide areas of applications [49], for example, new fractional order system models, electrical circuits theory – fractances, capacitor theory, etc.

A fractional order dynamic system can be described by a fractional differential equation of the following form [187], [188], [220]:

(1.13)

Numbered Display Equation

where inline ; ak (k = 0, … n), bk (k = 0, … m) are constants; and α k (k = 0, … n), β k (k = 0, … m) are arbitrary real numbers.

Without loss of generality, we can assume that α n>α n−1>…>α 0, and β m>β m−1>…>β 0.

To obtain a discrete model of the fractional order system (1.13), we have to use discrete approximations of the fractional order integro-differential operators and then we obtain a general expression for the discrete transfer function of the controlled system [220]

(1.14) Numbered Display Equation

where (ω (z−1)) denotes the discrete equivalent of the Laplace operator s, expressed as a function of the complex variable z or the backward shift operator z−1.

The fractional order linear time-invariant system can also be represented by the following state-space model

(1.15) Numbered Display Equation

where inline , inline and inline are the state, input and output vectors of the system and inline , inline , inline , q is the fractional commensurate order [161].

1.1.4 Stability of LTI Fractional Order Systems

It is known from the theory of stability that a linear time-invariant (LTI) system is stable if the roots of the characteristic polynomial are negative or have negative real parts if they are complex conjugate. It means that they are located on the left half of the complex plane. In the fractional order LTI case, the stability is different from the integer one. An interesting notion is that a stable fractional order system may have roots in the right half of complex plane (see Figure 1.1). It has been shown that system (1.15) is stable if the following condition is satisfied [153]

Figure 1.1 Stability region of LTI fractional order systems with order inline

c01f001

(1.16) Numbered Display Equation

where 0<q<1 and eig(A) denotes the eigenvalues of matrix A.

Matignon’s stability theorem says [153]: The fractional transfer function G(s) = Z(s)/P(s) is stable if and only if the following condition is satisfied in σ-plane:

(1.17) Numbered Display Equation

where σ := sq. When σ = 0 is a single root of P(s), the system cannot be stable. For q = 1, this is the classical theorem of pole location in the complex plane: no pole is in the closed right half plane of the first Riemann sheet.

Generally, consider the following commensurate fractional order system in the form:

(1.18) Numbered Display Equation

where 0<q<1 and inline . The equilibrium points of system (1.18) are calculated via solving the following equation

(1.19) Numbered Display Equation

The equilibrium points are asymptotically stable if all the eigenvalues inline of the Jacobian matrix inline , evaluated at the equilibrium, satisfy the following condition:

(1.20) numbered Display Equation

Figure 1.1 shows the stable and unstable regions of the complex plane for such a case.

Figure 1.2 Classic control system and its ideal Bode plot

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1.2 Fractional Order Controls

1.2.1 Why Fractional Order Control?

Using the notion of fractional order may be a step closer to the real world because the real processes are generally or most likely fractional [18]. However, for many of them, the fractionality may be very small. As said, a typical example of a non-integer (fractional) order system is the voltage-current relation of a semi-infinite lossy RC line. In theory, the control systems can include both the fractional order dynamic system or plant to be controlled and the fractional order controller. However, in control practice, it is more common to consider the fractional order controller. This is due to the fact that the plant model may have already been obtained as an integer order model in classical sense. In most cases, our objective is to apply the fractional order control (FOC) to enhance the system control performance. For example, the proportional integral derivative (PID) controllers, which have been dominating industrial controllers, have been modified using the notion of fractional order integrator and differentiator. It is shown that the extra degree of freedom from the use of fractional order integrator and differentiator made it possible to further improve the performance of traditional PID controllers [161]. Therefore, in this section, we will concentrate on this scenario–the controller being fractional order.

1.2.2 Basic Fractional Order Control Actions

The applications using fractional calculus have been attracting more and more attentions in the past few decades. These mathematical phenomena describe a real object more accurately than the classical integer order methods. As pointed out in [50], clearly, for closed-loop control systems, there are four situations. They are (1) IO (integer order) plant with IO controller; (2) IO plant with FO (fractional order) controller; (3) FO plant with IO controller and (4) FO plant with FO controller. From a control engineering point of view, doing something better is the major concern. Existing evidence confirm that the best fractional order controller can outperform the best integer order controller. It has also been answered in the literature why one should consider fractional order control even when the integer (high) order control works comparatively well [156, 159]. Fractional order PID controller tuning has reached a mature state of practical use. Since (integer order) PID control dominates the industry, we believe FOPID will gain increasing impact and wide acceptance. Furthermore, we also believe that based on some real-world examples, fractional order control is ubiquitous when the dynamic system is of distributed parameter nature [50].

1.2.3 A Historical Review of Fractional Order Controls

In [29], Bode mentioned, maybe the first time and in a comprehensive way, the interest of considering a fractional integro-differential operator in a feedback loop without using the term fractional. After that, Manabe [150] introduced the frequency and transient responses of the fractional order integral and its application to control systems. As a further step in automatic control, the tilted integral derivative (TID) controller was proposed in a patent [145] by Lurie, and Oustaloup proposed the CRONE (Commande Robuste d'Ordre Non Entier) method with three generations [176], [178], [182], over the traditional PID controller for the control of dynamic systems to achieve better performance. A generalization of the traditional integral order PID controller was proposed by Podlubny, e.g. the PIλDμ controller with an integrator of order λ and a differentiator of order μ [190]. Extending the classical lead-lag compensator to the fractional order case was studied in [193].

More early attempts to apply fractional calculus to systems control can be found in [15], [18], [71], [152], [175], [191], [199]. In this section, four representative fractional order controllers in the literature will be briefly introduced, namely, TID controller, CRONE controller, the PIλDμ controller and fractional lead-lag compensator.

1.2.3.1 TID Controller

In [144], a feedback control system using a PID controller is provided, wherein the proportional component of the PID controller is replaced with a tilted component having a transfer function inline . The resulting transfer function of the entire system with this TID more closely approximates an optimal transfer function, thereby achieving improved feedback controller. Further, as compared to conventional PID controllers, this TID controller allows for simpler tuning, better disturbance rejection ratio, and smaller effects of plant parameter variations on closed-loop response.

A. Basic Motivations

The motivation for this TID control is from the consideration of the so-called theoretically optimal loop response due to Bode. Consider the conventional feedback control system block diagram in Figure 1.2(a) where C is the feedback controller, yr is the reference input signal, e is the control error signal, u and y are input and output signals respectively. In Figure 1.2(a), the additive disturbance is denoted by v. The major goals of the feedback control system are to minimize the effect of disturbances at the output of the system, and to minimize sensitivity of the closed-loop response to plant parameter variations. To satisfy these requirements, the feedback of the system, properly weighted in frequency, must be maximized. These constraints uniquely define the optimal transfer function for the feedback loop. The purpose of the controller of the feedback system is to implement a loop response reasonably close to the optimal one. A commonly used controller employed in feedback control systems is a PID controller. In fact, a PID controller provides varying degrees of gain and phase shift of the signal according to the frequency contents. The conventional PID controller transfer function typically has two real zeros. Typically, the P-term dominates near fc, the D-term dominates at frequencies over 4fc, and the I-term dominates at frequencies up to fc /4, where fc is the crossover frequency at which the loop gain is 0 dB as shown in Figure 1.2(b).

Referring to Figure 1.2(b), a theoretically optimal loop response has been determined by Bode. For the purpose of industrial control, a simplified suboptimal Bode loop response can be employed. The suboptimal response is illustrated in Figure 1.2(b) by a dashed line. The slope of this suboptimal gain response is about −10 dB/octave. The transcendental loop transfer function which characterizes the suboptimal response can be closely approximated by a rational function. As can be seen from Figure 1.2(b), rather sharp corners occur at the sides of the Bode step. Any smoothing of the corners, especially the left one, caused by an improper or inaccurate rational function approximation, reduces the available feedback, resulting in reduced performance. A typical loop gain Bode plot of the system with a PID controller is also shown in Figure 1.2(b). When provided with the same stability margin and the same average loop gain as an optimal Bode controller, the crossover frequency fc of the PID controller is about one-half that of the optimal Bode loop response. The feedback at frequency fc/4 is about 10 dB lower than that of a simplified suboptimal Bode loop response. The conventional PID controllers in common use when applied to a great variety of plants, are easy to tune to provide robust and fairly good performance. However, the performance is not optimal as explained above.

The aim of TID is to provide an improved feedback loop controller having the advantages of the conventional PID controller, but providing a response which is closer to the theoretically optimal response.

B. Brief Introduction to TID Control

Similar to PID control, the TID control scheme is shown in Figure 1.3. where the the proportional compensating unit is replaced with a compensator having a transfer function characterized by inline or s−1/n. This compensator is herein referred to as a Tilt compensator, as it provides a feedback gain as a function of frequency which is tilted or shaped with respect to the gain/frequency of a conventional or positional compensation unit. The entire compensator is herein referred to as a Tilt-Integral-Derivative (TID) controller. For the Tilt controller, n is a nonzero real number, preferably between 2 and 3. Thus, unlike the conventional PID controller, wherein exponent coefficients of the transfer functions of the elements of the compensator are either 0, −1, or +1, the TID scheme exploits an exponent coefficient of −1/n. By replacing the conventional proportional component with the tilt component of the invention, an overall response is achieved which is closer to the theoretical optimal response determined by Bode as illustrated in Figure 1.2(b).

Figure 1.3 Block diagram of TID control scheme

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In Figure 1.3, R(s) is a pre-filter provided for a proper command signal pre-filtering which is commonly seen in practice. A preferred transfer function for the pre-filter is

Unnumbered Display Equation

Since the T-term eliminates static error, the coefficient of the I-term can be set to zero for many problems, thus simplifying controller tuning. A suggested tuning procedure for the TID controller is:

1. set KI = 0, KD = 0, and set the coefficient KT for the loop gain to be 0 dB at a desired crossover frequency fc;

2. set KD such that the phase stability margin at the crossover frequency is about5 degrees larger than desired; and

3. set KI = 0.25KTf(1−1/n)c.

Taking n = 1/3 as an example, the transfer function 1/s¹/³ can be approximated by a transfer function having alternating real poles and zeros in a complex plane representation. Three poles and three zeros per decade generally suffice to achieve the phase error of less than 1 degree and the amplitude error of less than 0.1 db which is given by

Unnumbered Display Equation

Enter the coefficients for the above approximated transfer function T6/6(s) for 1/s¹/³ into CtrlLAB© [237], three mouse clicks give the Bode plot, the Nichols chart and the root locus as shown in Figure 1.4.

Figure 1.4 Frequency responses of transfer function T6/6(s)

c01f004

From the Bode plot, we can see that T6/6(s) is a good approximation for 1/s¹/³ in both magnitude and in particular the phase (constant phase angle). The vertical line in the Nichols chart Figure 1.4(b), is a desired robustness property for controller design.

In TID patent [144], an analog circuit using op-amps plus capacitors and resistors is introduced with a detailed component list which is useful in some cases where the computing power to implement 1/s¹/n digitally is not possible. An example is given in [144] to illustrate the benefits of TID over conventional PID in both time and frequency domains.

1.2.3.2 CRONE Controller

The CRONE control was proposed by Oustaloup in pursuing fractal robustness [181], [183]. CRONE is a French abbreviation for "Contrôle Robuste d’Ordre Non Entier" (which means non-integer order robust control). In this section, we shall follow the basic concept of fractal robustness, which motivated the CRONE control, and then mainly focus on the second generation CRONE control scheme and its synthesis based on the desired frequency template which leads to fractional transmittance [174], [180].

A. Fractal Robustness

In [177], fractal robustness is used to describe the following two characteristics: the iso-damping and the vertical sliding form of the frequency template in the Nichols chart. This desired robustness motivated the use of fractional order controller in classical control systems to enhance their performance.

1. Iso-damping lines. Consider the characteristic equation

(1.21) Numbered Display Equation

where τ is a constant. The two poles are given by

(1.22) Numbered Display Equation

with 1<α <2. The poles are complex and conjugated, and form a center angle inline with inline as shown in Figure 1.5(a). Clearly, the poles move at a constant angle (fixed by the order α) when τ varies. The robustness in s plane is then illustrated by two half-straight lines which form the same angle inline in relation to the real axis and are called iso-damping half-straight lines.

The natural frequency and the damping ratio are directly deducible from the poles, through their modulus 1/τ and the half-center angle inline as follows:

(1.23) Numbered Display Equation

and

(1.24) Numbered Display Equation

It can be clearly seen that the damping ratio inline is exclusively a function of the fractionality order α, thus allowing the introduction of the notion of robust oscillatory mode.

2. Frequency template. With a unit negative feedback, the forward path transfer function, or open-loop transmittance, for the characteristic equation (1.21) is

(1.25) Numbered Display Equation

which is the transmittance of a non-integer integrator in which ω u = 1/τ denotes the unit gain (or transitional) frequency.

As inline with 1<α <2, the Nichols chart of β (jw) is a vertical straight line between inline and inline . This is illustrated in Figure 1.5(b). When τ, the system parameter, changes, the vertical straight line shown in Figure 1.5(b) slides. Such a vertical displacement ensures a constant phase margin inline , and thus correspondingly a constant damping ratio in the time domain.

In controller design, the objective is to achieve such a similar frequency behavior, in a medium frequency range around ω u, knowing that the closed-loop dynamic behavior is exclusively linked to the open-loop behavior around ω u. Therefore, the ideal controller design comprises:

1. An open-loop Nichols locus which forms a vertical straight line segment around ω u for the nominal parametric state of the plant, called the open-loop frequency template (or more simply the template) (Figure 1.5(b));

2. A sliding of the template on itself when there are parameter changes in the plant (assume that the parameter change will lead to gaining variations around ω u).

Synthesizing such a template defines the non-integer approach that the second generation CRONE control uses.

B. The Second Generation CRONE Control: Basic Concept

For a typical disturbed feedback control system as shown in Figure 1.2(a), its control performance is fully characterized by the sensitivity function inline , also known as the transmittance in regulation, or the complementary sensitivity function inline , also known as the transmittances in tracking, and we know that inline . It is practically true that given the open-loop behavior around the unit gain frequency, one can determine the dynamic behavior in closed loop. Therefore, we use the transmittance frequency template , β (s), as shown in Figure 1.5, to define the desired behavior of inline or inline . Let’s choose a template such