Chapter 4

LINEAR INTEGER ORDER SYSTEM CONTROL

BY FRACTIONAL PI-STATE FEEDBACK

Rachid Mansouri1, Maamar Bettayeb2,3, Chahira Boussalem1

and Ubaid M. Al-Saggaf 3

1L2CSP Laboratory, University Mouloud Mammeri of Tizi-Ouzou, Algeria 2Electrical & Computer Engineering Department, University of Sharjah, UAE 3 Center of Excellence in Intelligent Engineering Systems (CEIES),

King Abdulaziz University, Jeddah, KSA

Abstract

This book chapter deals with a linear non-fractional order system control by a PI-state feedback fractional-controller. The fractional aspect of the control law is introduced by a

The first one uses the augmented system method. In this case, the closed-loop

closed-loop, the Bode's ideal transfer function.

These two methods are then applied to stabilize an inverted pendulum-car system and the effectiveness and robustness of the proposed controllers are examined by experiments on a real system.

Keywords: Fractional calculus, Fractional systems, PI-state feedback, Pole placement, Inverted pendulum-cart system stabilization

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Introduction

The idea of fractional calculus has been known since the development of the regular integer order calculus, with the first reference being associated with Leibnitz and L'Hopital in 1695 where half order derivatives were mentioned. A detailed chronology of this notion is given in [23, 36]. The books [31, 41, 44] and more recently [18, 22, 33] constitute currently the main references of this theory. Non integer differentiation is currently considered among emerging thematics applied to many science and technology disciplines. All works undertaken on systems modeling by linear differential equations using the usual integer differentiation must be re-examined. From the automation point of view, the fundamental notions analysis and simulation of fractional systems and the fractional controllers design constitute highly promising fields of research.

In control theory, the non integer differentiation/integration has also proven its efficiency. Indeed, Oustaloup, [40] with the CRONE-controller (Commande Robuste d'Ordre Non Entier) and Podlubny, [42] with the -controller, involving the fractional-order integrator and differentiator, have shown the advantage of the fractional-controllers over the classical ones. Fractional-order controllers have received a considerable attention in the last three decades (see [7, 12, 24, 33] for example). In fact, in principle, they provide more flexibility in the controller design, with respect to the integer order controllers, because they have more parameters to select.

However, almost all fractional controller design methods use the transfer function representation because of the simplicity of the mathematical manipulations. There is still a lack of strategies for introducing fractional dynamics in state space methods. The state space representation of fractional order systems is introduced in [13, 19, 43, 49]. It has been exploited in order to analyze performance of fractional order systems. Indeed, the stability of the fractional order system has been investigated [28, 30]. Further, the controllability and the observability properties have been defined and some algebraic criteria of these two properties have been derived [8, 29, 51]. Recently, some new results on the minimal state space realization for fractional order systems [47, 48] and approximation of fractional order state space SISO and MIMO models in both commensurate and non-commensurate cases are derived [25-27].

Control systems can include both the fractional order system to be controlled and the fractional order controller. However, in practice, it is more common to consider only the controller to be of fractional order. This is due to the fact that the plant model may have been already obtained as an integer order model in the classical sense. In state space representation, the introduction of the concept of fractional differentiation is not obvious because it does not appear explicitly in the model. As in the transfer function representation, the fractional aspect will be introduced by the control law. However, in state space, the control law is usually used as a static state feedback that does not allow the introduction of the fractional aspect. In [34, 46], the authors present a fractional order state space control strategy for integer order systems. It is based on rearranging integer order system equations into a set of fractional (rational) order equations using system augmentation. By doing so, fractional order dynamics can be introduced in the controlled system by using traditional state-space methods for controller design. System augmentation is also used in [14, 15] to study the stabilization

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problem of commensurate fractional order systems by a pseudo-state feedback. However, the non-integer order must be rational.

Pole placement via state feedback has been known for several decades [35]. Recently, state-derivative feedback has been proposed [1-3, 18, 36] due to considerably low gain and fast response. In this chapter, new approach is presented to design the fractional PI-state feedback gain, two solutions are then proposed. In the first one, the integer PI-state feedback solution proposed in [50] is generalized to the fractional PI-state feedback. We show that we find the same solution but also the same problems as the solution proposed in [46] by using system augmentation principle. Our second approach [10] provides a very simple method to achieve the gain vectors of the fractional PI-state feedback. The desired closed-loop characteristic polynomial is composed of two polynomials: The first one is integer and

40] (phase flatness of the open-loop) is used. These two methods are then applied to stabilize an inverted pendulum-car system.

This chapter contains six sections including the introduction. In Section 2, some basic definitions are recalled. These definitions include the fractional integral operation, the Bode's ideal transfer function and the stability condition of fractional order systems. Pole placement by integer PI-state feedback algorithm is also presented in this section. The main results are presented in Sections 3 and 4. In Section 3, the pole placement method by integer PI-state feedback is generalized to the fractional one. Limits of this method are also investigated and presented through a first order system. Section 4 contains the second method which is more general and simpler to implement. After presenting the inverted pendulum-car system in Section 5, those two methods are used in Section 6 to stabilize it. Experimental results are also presented in this section.

1. Basic Definitions and Preliminaries

1.1. Fractional Integral Operation

is an integer number. The generalization of the Cauchy formula (1) to a non-integer number is given by [20]

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Oustaloup [38] calls the weight function ( ) the forgetting factor ("le facteur d'oubli").

1.2. Bode's Ideal Transfer Function

In his study on the design of feedback amplifiers, Bode [10] suggested an ideal shape of the open-loop transfer function of the form

the slope of the magnitude curve, on log-log scale, and may assume integer as well as non-

Let us now consider the unity feedback system represented in Figure (1) with Bode's ideal transfer function ( ) inserted in the forward path. This choice of ( ) gives a closed-loop system with the desirable property of being insensitive to gain changes, also called the "iso-

Figure 1. Fractional-order control system with Bode's ideal transfer function.

The closed-loop system of Figure (1) is given by

It exhibits important properties such as infinite gain margin and constant phase margin (dependent only on ). Thus, this closed-loop system is robust to process gain variations and the

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the overshoot (dependent on )) and the settling time (dependent on ), respectively [38, 40].

1.3. Stability of the Fractional Order Systems

The definition of stability adopted here is the Bounded Input Bounded Output (BIBO) stability, also called external stability. For commensurate non-integer order systems, the stability condition is, as for the integer system, that the roots of the characteristic equation lie in the left-hand s-plane.

In practice, checking the stability condition by calculating the roots of the characteristic equation is very difficult due to the complexity of their calculation. Instead of using the roots

root .

Remark: The commensurability condition of the fractional order is a necessary condition. Indeed, when this condition is not verified, the study of fractional order system stability based on the roots of its characteristic polynomial only is not feasible.

1.4. Pole Placement by Integer PI-State Feedback

State feedback is a well established tool for the stabilization and control of dynamic systems, and it is widely used in the field of control engineering.

The idea of using state-PID feedback is relatively new, and so far only few applications can be found in the literature. In [1-3, 18], the motivation for using state derivative feedback instead of conventional state feedback is given by the easy implementation in mechanical applications where accelerometers and velocity sensors are used for measuring the system motion.

Thus, the accelerations and velocities are the sensed variables, as opposed to the displacement. Typical applications are in the vibration control of mechanical systems [2, 3, 18]. The use of acceleration feedback in vibration suppression problems has also been discussed at length in [37].

The application of state derivative feedback to vibration problems has been discussed from the perspective of robust control in [21]. In what follows, we recall the pole placement method in the integer PI-state feedback case, see [50] for more details.

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to achieve a desired closed-loop characteristic polynomial. The block diagram of Figure (2) represents a LTI system with PI-state feedback.

Figure 2. Block diagram representation of a linear system with PI-state feedback.

The corresponding characteristic polynomial is given by

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( )

where

( ) ( )

{( )

( ) having the controllable canonical form, the characteristic polynomial of the

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2. Pole Placement Based Design of the Fractional Pi-State

Feedback

The generalization of the PI-state feedback involves replacing the integer order integration by a fractional order operator. Figure (3) shows the block diagram of an integer system controlled by the fractional PI-state feedback. The generalization is given by the following result.

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Figure 3. Block diagram of a fractional PI-state feedback.

Theorem 2. Consider a controllable LTI integer order SISO system described by Equation (9) and let the control law

being the linear transformation that gives the controllable canonical form.

Proof. The characteristic polynomial of the integer model of Equation (9) is given by (11). Since the model (9) is controllable, it can be rewritten as (12). The control law (23) becomes

Laplace transform of Equation (28) yields

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which can be written as

Matrix ( ) has the general form

where

( )

and

̃( )

̃( )

{̃ ( )

The characteristic polynomial of the closed-loop system is then

Let ( ) the polynomial that we want to impose to the closed-loop

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the desired polynomial (34). The problem is how one can find the values of these coefficients from the poles imposed to the desired polynomial, as in the integer case. The polynomial

(34) being fractional, the number and the values of its roots depend on the value of .

constraints to be checked when one chooses the poles of the integer polynomial (39). In addition, to ensure the stability of the closed-loop system, poles must verify the Matignon's stability conditions given in Theorem 1. To illustrate the solution, consider the simple

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The computation of the fractional PI-state feedback is summarized in the following algorithm.

Consider the state space model of an integer order system for which the characteristic polynomial is written as

is a row vector

is the controllability matrix

is a unit vector

Step 3. Write the integer order polynomial, representing the closed-loop characteristic polynomial, in the form

constraints of Step , and with respect the Matignon's stability conditions.

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3. Bode's Ideal Transfer Function Based Design of Fractional Pi-State Feedback

We presented, in Section 2, the generalization of the integer PI-state feedback. The pole placement design method, although it can be used to determine the feedback gains and , is not practical because the non-integer order must be rational and its value is not chosen according to a specified objective. In addition, the poles of the corresponding integer polynomial are not arbitrary since they must verify constraints because some of the polynomial coefficients are zero. In this section, another method based on the Bode's ideal transfer function is proposed.

Theorem 3. Let's consider a controllable LTI integer order SISO system described by Equation (9) and let the control law given by Equation (23). Vectors and that can arbitrarily place the closed-loop characteristic polynomial written in the form

are given by

being the linear transformation that gives the controllable canonical form.

Proof. The closed-loop characteristic polynomial of the state space model (12) using the

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is chosen to impose the dynamics of the closed-loop step response and the non-integer order

other hand, and taking into account Equation (48), gives

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(54)

and

The pole placement of the closed-loop for the original system (9) is achieved by using the gain vectors

This completes the proof.

The following algorithm, for the computation of the fractional PI-state feedback, results directly from the proof of the theorem.

is a row vector

is the controllability matrix

is a unit vector

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5. Inverted Pendulum-Cart System Modeling and Swing-Up

Control

The inverted pendulum problem is a challenging problem in control theory and has been studied extensively in the control literature [11, 16, 17, 45]. It is a well established benchmark problem that provides many challenging issues in control design. The system is nonlinear, unstable, non-minimum phase and underactuated. The dynamics are quite similar to twowheeled mobile robots, flexible link robot, biped robot limbs, missiles and so on. Because of their nonlinear nature, pendulums have maintained their usefulness and they are now used to illustrate many of the ideas emerging in the field of non-linear control. The control of the inverted pendulum can be divided into three aspects. The first aspect that is widely researched is the swing-up control of the inverted pendulum. The second aspect is the stabilization of the inverted pendulum. The third aspect is the tracking control of the inverted pendulum. In practice, stabilization and tracking control are more useful for applications.

The pendulum-cart system is shown in Figure (4) and consists of

A pendulum hinged in the cart by means of ball bearings rotating freely in the plane containing this line,

A cart driving device containing a DC motor, a DC amplifier and a pulley-belt transmission system.

The control input is the supply voltage of the DC motor and the outputs are the cart position and the pendulum angle .

The complete model describing the dynamic behaviour of the pendulum-cart system is given by [4, 5]

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Figure 4. Inverted pendulum-cart system.

To avoid complicating the model of the pendulum-cart system, the DC motor dynamics

Swinging up an inverted pendulum will return it from the stable equilibrium point in position to an unstable equilibrium point in upright position or . The swing-up strategy used here is based on the Lyapunov stability theorem that allows the convergence of the system energy to its energy value in an upright position. The Lyapunov function used is the difference between the mechanical energy of the pendulum and the

to protect the DC motor against the control effort. Figure (5) shows the evolution of the cart

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Figure 5. Experimental results of the swing-up control.

5. Implementation of the Fractional PI-State Feedback to Stabilize the Inverted Pendulum-Cart System

To test the theoretical results elaborated in Sections and , the linearized model of the inverted pendulum-cart system around is used. It is given by

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velocity ̇ and the angular velocity of the pendulum ̇must be measured. These two last measurements, being not available on the real system, we have used a differentiator followed by a low-pass filter whose transfer functions are

(62)

Furthermore, The association between swing-up control and stabilization control requires adding a suitable switching law [5, 45]. In our case, the switching is applied when the absolute value of the angle is close to .

5.1. Implementation of the Fractional PI-State Feedback Obtained Using the First Method

polynomial corresponding to the closed-loop characteristic polynomial. The integer polynomial is given by

zero coefficient is that corresponding to . So, the chosen poles must verify this constraint

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Figure (6) shows the experimental results obtained. This figure shows the effectiveness of the fractional PI-state feedback control on the experimental test bench. Indeed, it maintains

Figure 6. Experimental results swing-up + stabilization control using the first method.

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is obtained by adding a small mass of aluminum on the cart just after the end of the swinging up step). Figure (8) shows the results obtained when the pendulum is subjected to disturbance in the form of forces applied on it (by hitting the pendulum with a pen at different times). Figures (7) and (8) show the robustness of the stabilization by the fractional PI-state

5.2. Implementation of the Fractional PI-State Feedback Obtained by Using the Second Method

For the purpose of fractional order PI-state feedback control design, the linearized model of the inverted pendulum-cart system (Equation 59) is considered again. The linear model

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Figure (9) shows the experimental results obtained. This figure shows the effectiveness of the fractional PI-state feedback control on the experimental test bench. Indeed, it maintains

Figure 9. Swing-up and fractional order PI-state feedback control experimental results, using the second method.

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