Control Techniques for a Class of Fractional Order Systems

: The paper discusses several control techniques for a class of systems described by fractional order equations. The paper presents the unit frequency criteria that ensure the closed loop control for: Fractional Order Linear Systems, Fractional Order Linear Systems with nonlinear components, Time Delay Fractional Order Linear Systems, Time Delay Fractional Order Linear Systems with nonlinear components. The stability criterion is proposed for the systems composed of fractional order subsystems. These techniques are used in two applications: Soft Exoskeleton Glove Control, studied as a nonlinear model with time delay and Disabled Man-Wheelchair model, ana-lysed as a fractional-order multi-system.


Introduction
In the last decade, the extension of the applications of the fractional order systems (FOS) has determined a corresponding increase of the control techniques, of the methods of investigation of the special requirements imposed by these systems. Enumerating these methods would be an extremely difficult operation. A summary of these efforts can be found in [1,2]. Seeking to capture certain aspects in this domain, we could identify some directions that mark this activity. The stability criteria of FOS are obtained both by conventional methods deriving from the study of transfer functions with fractional exponent [3][4][5] and from those derived from Lyapunov's second method [6][7][8][9][10]. Mittag-Leffler stability is studied by Lyapunov techniques in [11]. FOS models defined by Caputo derivatives are analyzed in [12,13]. Nonlinear FOS or linear FOS with nonlinear components are treated in [14][15][16][17][18][19]. A physical interpretation of fractional viscoelasticity based on the fractal structure of media is investigated in [20]. Nonlinear optimal control techniques are proposed in [21]. Other control and stability issues refer to delayed FOS models. Stability criteria are formulated in the form of algebraic criteria [22][23][24] or are derived by analytical techniques derived from the Lyapunov method [25][26][27][28]. The control of multi-system architectures defined by FOS models is discussed in [29][30][31]. Frequency criteria for the control of Linear FOS are presented in [32][33][34]. A comprehensive review of literature related to the industrial use and integration of FOPID control is presented in [35]. FPGA implementation techniques of Fractional Order Systems is discussed in [36] and analog-hardware implementation of a FOPID for a DC motor [37] is emphasized. Memristor-based implementation with neuromorphic circuits which provide guidelines for physical realization of FOPID controllers, and a discrete implementation are presented in [38,39] respectively. Motivated by the above discussion, our main objective in this paper is to develop new stability criteria for FOS models via the approach in which the control laws are  9, defined by frequency criteria. The main contribution of this paper, compared with the existing results, are listed as follows: • Frequency criterion for linear FOS is proposed. Lyapunov techniques and the methods that derive from Yakubovici-Kalman-Popov Lemma [40] are used and the criteria that ensure asymptotic stability of the closed loop system are inferred. The criterion is extended to linear FOS with nonlinear components. • Frequency criterion for time delay FOS is proposed using an approximate model based on Grunwald-Letnikov formula. The result is extended to the systems with nonlinear components. • Stability criterion is proposed for systems composed of fractional order subsystems.

•
The proposed methods have been implemented on two major applications: Soft Exoskeleton Glove Control that is studied as a nonlinear FOS model with time delay and Disabled Man-Wheelchair system that is studied as a fractional-order multi-system.
The paper is structured as follows: Section 2 presents mathematical preliminaries, and the control techniques proposed in this paper, Section 3 treats two applications: Soft Exoskeleton Glove Control and Disabled Man-Wheelchair system control, Section 4 is dedicated to Discussion and Section 5 presents Conclusions.

Control Techniques
Notation 1. Let ∈ × be a real matrix. We denote the eigenvalues of by ( ). We denote by ( ) , ( ) , the minimum eigenvalue and maximum eigenvalue of , respectively. Write We denote by the complex number, = −1.
Consider the following control law where the controller gain = + is a positive constant that verifies the conditions and is a positive constant.
By evaluating (15) along the solutions of (7)-(10), yields where = − . This inequality can be rewritten as Then by (10) and (12) it follows that Substituting this result into (17) results in By employing Yakubovici-Kalman-Popov (YKP) Lemma [40] and the condition (11), it follows that Now, substituting this result into (19) and considering the control law (10), it yields or,

Proof. The Lyapunov function
Therefore, applying inequalities (33), (35) yields where * = The inequality (41) can be rewritten as where is given by (26). □ Remark 1. The result of Theorem 3 can be easily extended to systems with nonlinear components, where ( ) verifies the condition (26). In this case, the frequential condition (38) is verified for * + ( ) , where , , are defined in Section 2.2.2.

Systems Composed of Linear Fractional Order Subsystems
Consider the linear systems where the exponent, is a ( × ) constant matrix, ∈ ∁ In the composite system, a number of components can be -fractional order stable systems, Consider the Lyapunov function associated with each system , ( : → ), From (4) and Lemma 1 (6), yields or Consider the following composite system where is a × inter-connection matrix that satisfies the following condition where the following inequalities were used [29]: Consider that the composite system state vector is Theorem 6. The composite system (45), (51) which satisfies the following condition Proof. Consider the following vector Lyapunov function, ( : ∑ → ) where , = 1,2, . . , are defined by (48), ( ) is a continuously differentiable vector function and and is a positive constant. The fractional derivation of (14) is and according with (48), (50), and Lemma I (6), yields By employing inequality (52) one derives that or, From (59), it follows that If the condition (56) is satisfied, the system (65) is asymptotically stable, such that It can be concluded that lim → ( ( )) = 0 , = 1,2, . . , or, lim → ( ( )) = 0 , = 1,2, . . , QED The control strategy proposed by Theorem 4 based on the Lyapunov vector function (58) allows the use of ( × ) matrix operators instead of (∑ × ∑ ) operators as used in [38,39], which considerably simplifies the calculation and implementation effort. □

Control Systems-Conclusions
The previous sections discussed the control laws for several classes of models described by fractional-order equations: Fractional Order Linear Systems, Fractional Order Linear Systems with nonlinear components, Time Delay Fractional Order Linear Systems, Time Delay Fractional Order Linear Systems with nonlinear components. The control law was determined for these systems as, ( ) = − ( ) where the controller gain = + and verifies the condition ≤ 1. The component satisfies the frequency condition: ( ( )) ≥ −√ , ∀ ∈ (−∞, +∞), where ( ) is determined by (13), (28), (36), for each class of systems. This inequality can be interpreted as the Nyquist criterion for the critical point −√ .

Soft Exoskeleton Glove Control
An illustrative example of FOLS models is the model of a biomechanical system of a glove exoskeleton used to restore human functions for people with disabilities or who have suffered various traumas. Figure 1 shows the architecture of such a system that represents a complex structure of muscle tissues combined with elements of mechanical structure. An advanced conceptual analysis of the dynamics of these systems indicates the use of fractional calculus in Maxwell's classic stress-strain models [41,42].
The output of the system ( ) is evaluated by direct measuring of the angular position = and by estimation of fractional variables , by using the Gr nwald-Letnikov's formula, The nonlinear component ( ) verifies the inequality (26) with = 2 × 10 . The dynamic system is described by the following parameters [34]: the equivalent inertia is = 14 × 10 . , the viscous and elastic coefficients of the joint tissues throughout phalange musculoskeletal system and glove are = 0.408 × 10 . . , = 0.27 × 10 .
, respectively and the damping coefficient is = 0.324 × 10 . .   MATLAB/SIMULINK methods are used for the simulation [4,10]. The variable trajectories, the physical significance variables as angular position, and the estimated variables by Mittag-Leffler techniques, , , are shown in Figure 4. The asymptotic stability of all variables, measurable physical significance, and estimated virtual variables is well highlighted. Note the evolution of the angular variable , the variable that defines the main coordinate of movement, that satisfies the parameters: the settling time and the peak time verifies the condition < 0.5 for = 5 with an overshoot < 0.05% [40].  is selected as = 3.5 to satisfy the constraint (37) for = 0.25. The factor is evaluated, =3.2, and the parameter * is estimated as * = 1.6 for ∈ [0.1, 0.5] . The frequency condition (38) is verified ( Figure  3) and the condition (39) is satisfied. The system trajectories are shown in Figure 5. As in the previous example, we remark the asymptotic stability of all variables, measurable physical significance, and estimated virtual variables. Regarding the angular variable , the transient response satisfies the parameters: < 0.5 for = 6 with an overshoot < 0.08%.

Disabled Man-Wheelchair System Control
One of the most interesting control applications of some systems defined by fractional order models is the one referring to the control exercised by a disabled person, with health problems regarding locomotor disabilities or cerebral deficiency. Such an example may occur in a person with Parkinson's who drives a wheelchair ( Figure 6). This system can be considered as a composite system with two sub-systems: the human operator and the wheelchair. The existence of a handicap of the human operator determines a wheelchair driving with very low performances. For this reason, it is necessary to introduce a controller to improve the control of the movement on the imposed trajectories, (Figure 7). The dynamics of human operator are better characterized by fractional models that more correctly express the control strategy based on memorizing some observations from the history of evolution. The fractional order model is [43][44][45], where is the human state, , are human time constants, is the human gain (see Appendix A), defines the human control toward the target, the desired evolution expressed by a desired wheelchair orientation.
where , a (4× 1) matrix, designates the influence of the components of on human behavior. The human control acts through a negative feedback determined by the information received from the driven system (the wheelchair), where is (4× 1) interconnection matrix between the two systems (Appendix A). The model of the wheelchair is a classical wheelchair model [46] with the control of direction done by the control of the main wheel velocities and detremined by the joystick, The model represents a decoupled drive system, with symmetrical, electrical, and mechanical structure. This symmetry allows to use the following mathematical model The dynamic model is The system (90) The controller implements the control law for the composite system where , a (3 × 1) matrix, designates the human control components in the global control law (Appendix A).
As proposed in Section 2.2.4, we define the Lyapunov function of the composite system The system trajectories, human operator, and wheelchair, for a desired target defined by = π/2 rad are shown in Figure 8. The good quality of motion can be easily remarked. We remark the asymptotic stability of all variables, measurable physical significance, and estimated virtual variables, . , . The same behavior is noticed in the parameters that define the human operator, , , . Regarding the control angular variable ω, the transient response has an overshoot δ < 0.01%.

Discussion
The control techniques presented by the literature are based on algebraic criteria, matrix inequalities and frequency criteria. A selection of these criteria are presented in Table  1

Conclusions
In this paper, Lyapunov techniques are used to formulate frequency criteria for several classes of fractional order systems. Asymptotic stability for the closed loop systems is studied and frequency and algebraic conditions are inferred. FOS linear models, or with nonlinear components are analyzed. A class of delay time FOS systems is analyzed by using an approximate model obtained by the Grunwald-Letnikov formula. Algebraic stability criterion is proposed for systems composed of linear fractional order subsystems. The obtained results were applied on two important applications: the control of a Soft Exoskeleton Glove and the control of a wheelchair of a disabled person.