Fractional Order Controller Strategy for Nonlinear Dynamic Systems H. Delavari*, R. Ghaderi*, A. Ranjbar N. 1**, S. Momani*** *Noushirvani University of Technology, Faculty of Electrical and Computer Engineering, P.O. Box 47135-484, Babol, Iran, ([email protected] ), ** Golestan University, Gorgan P.O. Box 386, Iran ([email protected] ) *** Department of Mathematics, Mutah University, P.O. Box: 7, Al-Karak, Jordan Abstract: In this paper, a fractional order controller is proposed for nonlinear system. In this method the fractional controller converts the system with integer derivatives into a system with desired fractional derivatives in order to increases the degree of freedom of the stability. Preliminary fractional order controller is designed for general form of the nonlinear systems, and then the performance of the controller is investigated for two nonlinear dynamic case studies. The stability of the closed loop system is also verified in presence of the proposed controller. At first this controller is designed for a robot system and thereafter for a twin-tank model. Then numerical simulations are presented to show the effectiveness of the proposed controller. Keywords: Fractional Derivative, Fractional Control, Stability, Robotic System, Twin-Tank. 1. INTRODUCTION Fractional calculus is an old mathematical topic since 17th century. Although it has a long history, its applications to physics and engineering are just a recent focus of interest. A fractional-order controller for stabilizing the unstable fixed points of an unstable open-loop system is proposed in (Tavazoei et al., 2008). In (Calderón et al., 2006) several alternative methods for the control of power electronic buck converters applying fractional order control (FOC) are presented. Many systems are known to display fractional order dynamics, such as chaotic equations in control engineering (Ge and Ou, 2008), earthquake oscillation (He, 1998), wave equation (Jafari and Momani, 2007), and Riccati differential equation (Odibat, and Momani, 2008; Cang et. al,. 2007). An algorithm to determine parameters of an active sliding mode controller in synchronizing to different chaotic systems has been studied in (Tavazoei and Haeri, 2007). In (Ahmad et al., 2004) the problem of chaos control of three types of fractional order systems using simple state feedback gains is studied. The control of a special class of Single Input Single Output (SISO) switched fractional order systems (SFOS) from the viewpoints of the Generalized Proportional Integral (GPI) feedback control approach and a sliding mode based Σ − Δ modulation implementation of an average model based designed feedback controller is addressed in (Sira-Ramirez et al., 2006). In the robotics context, feedback linearization has accomplished (Park et al., 2000). Using this method, all the coupling nonlinearities in the dynamical model of the nonlinear system are first compensated, transforming the nonlinear system into a linear one. Then, the controller is designed on the basis of the linear and decoupled plant. In (Ferreira et al., 2003), the implementation of fractional order algorithms in the position/force hybrid control of robotic manipulators is 1 studied. The signal propagation and the fractional-order dynamics during, the evolution of a genetic algorithm, for generating a robot manipulator trajectory are addressed in (Pires et al., 2003). In this paper, we propose a fractional-order controller to control nonlinear dynamic systems. Also, we show that the proposed controller can reduce the reaching time and error in tracking control. This paper is organized as follows. Section 2 includes basic concepts in fractional calculus. In Section 3, a fractional-order controller is proposed. The implementation of proposed controller for robotic system and twin-tank model are studied in section 3.1 and 3.2 respectively. Finally, concluding remarks are drawn in Section 4. 2. FRACTIONAL CALCULUS Fractional calculus is a mathematical topic with more than 300-year history, but the application in physics and engineering has been recently attracted lots of attention. During past three centuries, this subject was with mathematicians, and only in last few years, this was pulled to several (applied) fields of engineering, science and economics. Perhaps the fractional calculus will be the calculus of twenty-first century. The fractional-order differentiator can be denoted by a general fundamental operator a Dtq as a generalization of the differential and integral operators, which is defined as follows (Calderón et al., 2006): q d dt q q a Dt 1 t ( d ) q a , R(q) 0 , R (q) 0 , R(q) 0 Corresponding Author: [email protected] , Tel: +98 911 112 0971, Fax: +98 111 32 34 201 (1) where q is the fractional order which can be a complex number, the constant a is related to the initial conditions. There are two commonly used definitions for the general fractional differentiation and integration, i.e., the Grünwald– Letnikov (GL) and the Riemann Liouville (RL). The GL definition is as: (2) (t q ) h 1 q j q D f ( t ) lim ( 1) f ( t jh ) a t h 0 h q j j 0 Where . is a flooring-operator while the RL definition is given by: t (3) 1 dn f ( ) q d a Dt f (t ) (n q) dt n (t )q n 1 a For (n 1 q n) and ( x) is the well known Euler’s Gamma function. Also, there is another definition of fractional differintegral introduced by (Caputo, 1967). Caputo’s definition can be written as: t (4) 1 f ( n) ( ) q D f ( t ) d , n 1 q n a t (q n) (t )q n 1 a Fractional order differential equations are at least as stable as their integer orders counterparts, This is because; systems with memory are typically more stable than their memoryless alternatives (Ahmed et al, 2007), Consider the following autonomous commensurate fractional order system: (5) D q x f ( x) , Whilst q is the commensurate order of the fractional timederivative in the Riemann-Liouville sense, and 0 q 1 and x 2 R . The equilibrium points of system (5) are calculated when: (6) f (x ) 0 . These points are locally and asymptotically stable if all eigenvalues of Jacobian matrix J f / x , which are evaluated at the equilibrium points, satisfy the following condition (Matignon, 1996): (7) arg(eig ( J )) q / 2 . n 3. CONTROLLER DESIGN: Applying the proposed fractional order controller, a system of integer derivatives will be transformed to a system of fractional order. Consequently, the degree of achieving the stability will be increased (Tavazoei et al., 2008). Suppose there is a nonlinear dynamic system which the governing dynamical equations can be written as: (8) x2 k 1 (t ) x2 k (t ) x2 k (t ) f k ( x, t ) bk ( x)uk (t ), x0 x(t0 ) Where t R , k 1, 2,..., n x(t ) [ x1 (t ) x2 (t ) x3 (t )... x2n (t )]T R2n (9) x(t ) is the state vector, u(t ) R is the control action vector, x0 x(t0 ) is the arbitrary initial conditions given at initial time t0 , bk ( x) and fk ( x, t ) , k 1,2,..., n are the control gains and the nonlinear dynamics of the system respectively. The desired state variables are defined as: xd (t ) [ xd1 (t ) xd 2 (t ) xd 3 (t )... xd 2n (t )]T R2n (10) The tracking error e(t ) R2n can be defined as: (11) e(t ) x(t ) xd (t ) The control objective is to get the states x(t ) to track the specific states xd (t ) . In other meaning, it is required to drive the tracking error asymptotically to zero for any arbitrary initial conditions. The proposed fractional controller in (Tavazoei and Haeri, 2008) increases the boundary of the stability. The general form of the proposed controller is written as follows: x2k (t ) k D q ( x2k (t )) (12) uk (t ) bk1 ( x) f k ( x, t ) k D1 q ( xd 2k 1 (t )) K D1 q (e2k 1 (t )) k K0 D q (e2k 1 (t )) k 1 where K0 and K1 are positive constant. Meanwhile the rate of convergence will be adjusted by other parameter, i.e. k . Then replacing (12) in (8) results the tracking error of the closed loop system as: e2k 1 K1e2k 1 K0 e2k 1 0 (13) which represents an exponentially stable error dynamics, and the proposed fractional order controller guarantees the convergence of x(t ) to its desired value xd (t ) . In next section the proposed method is implemented for two case studies. 3.1. Implementation of Proposed Controller in robot manipulator A two-degree of freedom polar robot manipulator has one rotational and sliding joint in the (x,y) plane. Neglecting the gravity forces and normalizing unit mass and unit length of the arm, the mathematical model of the two-degree of freedom polar robot can be expressed as follows: x1 (t ) x2 (t ) (14) x1 (t ) M ( x1 (t ) a) x42 (t ) x2 (t ) ( m) u1 (t ) d1 (t ) x3 (t ) x4 (t ) 2 x1 (t ) M ( x1 (t ) a) x4 (t ) x2 (t ) x4 (t ) u2 (t ) d 2 (t ) J1 J 2 2 x ( t ) 1 M ( x (t ) a ) 2 1 where is the mass of the motional link, M is the payload, J1 and J2 are moments of inertia of the motional link with respect to the vertical axis through c and o respectively. The robot manipulator joints are driven according to the following desired trajectory: (15) xd 1 (0.5 )sin( t / 20) m, xd3 (t),x3 (t),e 3 (t),rad. 3 xd 3 (0.9 )sin( t / 20) rad x 3(t) e 3(t) 1 0 -1 -2 Applying the proposed controller yields: -3 x2 (t ) 1 D q ( x2 (t )) 2 x1 (t ) M ( x1 (t ) a ) x4 (t ) u1 (t ) ( m) 1 q 1 q 1 D ( xd 1 (t )) 1 K1 D (e1 (t )) K D q (e (t )) d (t ) 1 1 1 0 x d3(t) 2 0 5 10 15 20 25 30 35 40 30 35 40 Time(second) (16) (b) 0.4 x4 (t ) 2 D q ( x4 (t )) 2 x1 (t ) M ( x1 (t ) a ) 2 J1 J 2 x1 (t ) u2 (t ) x2 (t ) x4 (t ) 2 D1 q ( xd 3 (t )) 2 M ( x ( t ) a ) 1 1 q 2 K 3 D (e3 (t )) K D q (e (t )) d (t ) 3 2 2 2 u1 (t),N. 0.2 0 -0.2 -0.4 5 10 15 20 25 Time(second) (c) For the simulation studies, the following numerical values are employed for the robot model parameters: M=1.2kg, 1kg , J1=J2=1 kgm2, a=1m, and the initial conditions are [ x1 (0), x2 (0), x3 (0), x4 (0)]T [-0.25, -0.2, 0.6, 0.8]T . and the controller parameters are chosen from the author experience, 1 0.5 , k1 1 , k0 1 , 2 0.5 , k2 1 , k3 1 , q 0.8 . The simulation result for this controller has been shown in Fig. 1. After reaching time the actual trajectory response x1(t) is almost identical to the desired command x1d(t), the same results is noticed for x3(t)and x3d(t). Then using (16) in (14) results the tracking errors of the closed loop system: e1 K1e1 K0 e1 0 1.5 xd1 (t),x1 (t),e 1 (t),m. -0.5 5 10 15 20 25 30 35 40 Time(second) (d) Fig.1. Proposed fractional order controller (a): Tracking response of joint1 (b): Tracking response of joint2 (c): Control signal u1(t) (d): Control signal u2(t) A twin-tank system consists of two small tanks coupled by an orifice and a pump that supplies water to the first tank. The pump only increases the liquid level and is not responsible for pumping the water out of the tank. It is assumed that the back pressure created by the water-head does not affect the flow rate of the pump significantly. Tracking control of twintank has been studied already such as (Khan, et al., 2006; Delavari et al., 2007a, b). 2 1 0.5 0 -0.5 x d1(t) -1 x 1(t) -1.5 e 1(t) 5 0 3.2. Implementation of Proposed Controller in Twin-tank Model This represents an exponentially stable error dynamics. 0 0.5 (17) e3 K3e3 K2 e3 0 -2 u2 (t),N.m. 1 10 15 20 25 Time(second) (a) 30 35 40 A twin-connected tanks system is a nonlinear dynamic system which the governing dynamic equation can be written as: h1 U in U12 A h2 (U12 U out ) A where (18) U12 a12 2 g (h1 h2 ) ,For h1 h2 (19) U out a2 2 gh2 ,For h2 0 where h1 and h2 are the total water heads in Tank 1 and Tank 2, respectively, Uin is the inlet flow rate, U12 is the flow rate from Tank 1 to Tank 2, A is the cross-section area of Tank 1 and Tank 2, a12 is the area of the coupling orifice, a2 is the area of the outlet orifice and g is the gravitational constant. Moreover, Uin 0 means that pump can only force water into the tank. Uin U12 Tank 1 x2 (t ) 1 D q ( x2 (t )) (c12 2c2 2 ) 2 (27) z1 ) (c1c2 2)( z1 z2 z2 U in (2 A z2 c2 ) 1 q 1 q 1 D ( H ) 1 K1 D (e1 (t )) q K D ( e ( t )) 1 1 0 where e=x1(t)-xd(t)=h2(t)-H, is the tracking error. With k0 and k1 are positive constants, then using (27) in (24) results the tracking error of the closed loop system: (28) e1 k1e1 k0 e1 0 This represents an exponentially stable error dynamics. The height h2(t) will asymptotically converge to the desired value H. Hence the proposed controller guarantees the asymptotic convergence of the output to the desired amount. Let 1 0.5 , k1 1 , k0 1 , q 0.8 , simulation results for this controller have shown in Fig.3. h1 h2 g= 981 cm/sec.2 H= 7cm (0 cm3/sec<Uin<50 cm3/sec. Applying the control strategy for this system one can obtain the input control signal as: Uout Tank 2 Fig.2. Interconnected Twin-Tank schematic 7.5 Let: z1 h2 0, z2 h1 h2 0 (20) c1 a2 2 g A , c2 a12 2 g A Also the output of the coupled tanks is taken h2(t) , hence the dynamic model in (1) and (2) can be written as: z1 c1 z1 c2 z2 2 z2 z2 2 100 150 100 150 100 150 3 uin (t),cm /sec. 40 30 20 10 0 0 50 Time(second) (b) 2 1.5 2 1 0.5 (25) 0 z2 z2 z1 ) (26) -0.5 0 50 Time(second) (c) 2A z ) 2 Let parameters of system for this model are as: A=180 cm2 50 2.5 in f ( x) (c1 2c2 ) 2 (c1c2 2)( z1 b ( x ) (c 0 50 z1 ) y x1 It can be easily expressed as x f ( x) b( x)u where 2 5.5 (a) e(t),cm. 2 A z2 c2 U x 1 (t) 6 Time(second) y1 z1 Then the goal is to regulate the system output (h2(t)) to the desired value (H). Let: (22) z1 x1 x 2 c1 z1 c2 z2 The inverse mapping from x to z is given by: (23) x1 z1 z 2 2 ((c1 x1 x2 ) c2 ) Then the model in (21) can be expressed as: (24) x1 x2 2 x d1 (t) 6.5 5 (21) z2 c1 z1 2c2 z2 U in A x (c1 2c2 ) 2 (c1c2 2)( z1 2 xd1 (t),x1 (t),cm. 7 a2= 0.24 cm2 a21=0.5 cm2 Fig.3. (a): water level in Tank2 (b): The inflow rate into Tank1 (c): Tracking error using proposed fractional order controller. Simulation results of employing the proposed controller with +5% variations in system parameters have been shown in Table 1, 2 respectively. A fast tracking response is observed by employing the proposed controller in comparison with the response obtained by employing the feedback linearization. From Table 1, 2 it can be seen that the Reaching times (Rt) and the mean absolute value of errors (E) by employing the proposed fractional order controller is less than conventional feedback linearization. Table 1. Results of controller performances with 5% variation in parameters of the system Robot q 0.8 1 Rt1(sec.) 7.5121 8.1512 Rt2(sec.) 9.1121 10.4182 E1 0.0062 0.0071 E3 0.0152 0.0161 Table 2. Results of controller performances with 5% variation in parameters of the system Twin-Tanks q 0.75 1 Rt1(sec.) 55.12 65.34 E1 0.3404 0.4108 6. 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