Fractional Order Controller Design for Underactuated ...

Fractional Order Controller Design for Underactuated Mechanical Systems Abhaya Pal Singh ∗ , Faruk Kazi ∗ , N. M. Singh ∗ Vishwesh Vyawahare ∗∗ ∗

Veermata Jijabai Technological Institute, Mumbai, Electrical Engineering Department, INDIA (e-mail: [email protected], [email protected], [email protected]. ∗∗ Indian Institute of Technology (IIT), Bombay, Mumbai, (e-mail: [email protected]) Abstract: Many real dynamic systems are better characterized using a non-integer order dynamic model based on fractional calculus. 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 nature around us. In this paper we propose a Fractional Order Controller for an underactuated mechanical system named Pendulum on a Cart system (POAC). The proposed system has two degrees-of-freedom and one control input. The mathematical model is derived using Lagrangian method and the stabilization is achieved with constraints using Fractional Order Controller. Simulation are performed to validate the control law and compared with Integer Order Controller. Keywords: Fractional order controller (FOC), FOPID, IOC, PID, POAC, Underactuated mechanical system. 1. INTRODUCTION Underactuated mechanical systems have fewer actuators than degrees of freedom. Underactuated systems appear in a broad range of applications including Robotics, Aerospace Systems, Marine Systems, Flexible Systems, Mobile Systems, and Locomotive Systems. The underactuation property of underactuated systems is due to the following four reasons: i) dynamics of the system , ii) by design for reduction of the cost or some practical purposes , iii) actuator failure , iv) imposed artificially to create complex low-order nonlinear systems for the purpose of gaining insight in control of high-order underactuated systems. While many interesting techniques and results have been presented for underactuated systems, the control of these systems still remains an open problem [Faruk Kazi,Ravi N. Banavar,Philippe Mullhaupt,Dominique (2008), Ioannis Sarras, Faruk Kazi, Romeo Ortega and Ravi Banavar. (2010), YangQuan Chen, B. M. Vinagre, and Igor Podlubny. (2003), YangQuan Chen, Kevin L. Moore, Blas M. Vinagre, and Igor Podlubny. (2004), YangQuan Chen, Blas M. Vinagre, and Igor Podlubny. (2004), YangQuan Chen, Dingy Xue, and Huifang Dou. (2004), Jinsong Liang, YangQuan Chen, Blas M. Vinagre, and Igor Podlubny. (2004), Jinsong Liang, YangQuan Chen, Blas M. Vinagre, and Igor Podlubny. (2005) and Igor Podlubny. (1999)]. Important issues are: how can nonlinear control models be formulated for such systems and what are their controllability and stabilizability properties. These issues are thoroughly addressed in[Mahmut Reyhanoglu,Arjan van der Schaft,N. Harris McClamroch and Ilya Kol-

manovsky (1999)]. Numerous robotic tasks associated with underactuation have been studied in the literature[C. Chevallereau, J.W. Grizzle and C.H. Moog (2004)]. The Pendulum On a Cart is a very good example for control engineers to study performance of a control law for underactuated mechanical system. The POAC is a highly nonlinear system. This means that standard linear techniques cannot model the nonlinear dynamics of the system.Its underactuation property makes control more challenging. The POAC is an intriguing subject from the control point of view due to their intrinsic nonlinearity. Here,the problem is to balance a pole on a mobile platform that can move in only two directions, to the left or to the right.In this paper we apply Fractional Order PID Controller (FOPID) to balance the POAC system and comparing the results with Integer Order PID Controller. In order to obtain control surface, the inverted pendulum dynamics should be locally linearised. Moreover, application of these control techniques to a two or three stage inverted pendulum may result in a very critical design of control parameters and difficult stabilization[Ahmad Nor Kasruddin Bin Nasir (2006/2007)]. Fractional order control has been applied with success in rigid robots[Subhransu Padhee, Abhinav Gautam, Yaduvir Singh, and Gagandeep Kaur (2011)]. In [Manue F. Silva, J. A. Tenreiro Machado. (2006)] a fractional order position/force algorithm is proposed for manipulators and control algorithms have superior performance. Fractional order PID controller is also used in Genetic Algorithm

for the optimization purpose [Subhransu Padhee, Abhinav Gautam, Yaduvir Singh, and Gagandeep Kaur (2011)]. Temperature control arises in many engineering fields.For temperature control, usually, On/Off control, proportional control, and traditional PID controller have been widely used.By using FOPID controller as in [Hyo-Sung Ahn, Varsha Bhambhani and YangQuan Chen (2008)]a more accurate temperature profile tracking of the spatially distributed heat flow was achieved. FOC is a generalization of the IOC to a real or complex order[J. L. Adams, T. T. Hartley and C. F. Lorenzo (2006)]. Formally the real order generalization is introduced as follows:  dα  dtα    1 Dα =  Rt    dτ −α

if α > 0 if α = 0 if α < 0

Fig. 1. Inverted Pendulum on a Cart

0

where α ∈ < We have many concepts of FOC given by different people. Some of them for derivative are:a) Riemann-Liouville (RL):Zt

m

Dα f (t) =

1 d [ m dt Γ(m − α)

f (τ ) dτ ] (t − τ )(α+1−m)

(1)

0

m ∈ Z + , (m − 1) < α ≤ m b) Grunwald-Letnikov (GL):t−a h 1 X Γ(α + 1) (−1)m Dα f (t) = lim α f (t − mh) h→0 h m!Γ(α − m + 1) m=0 (2)

c) Caputo:-

1 D f (t) = [ Γ(m − α) α

Zt

f m (τ ) dτ ] (t − τ )(α+1−m)

2. MODELLING 2.1 Modelling of Inverted Pendulum On A Cart system (POAC):The inverted pendulum mounted on a cart as shown in Fig.(1), the defining nonlinear equations can be derived as follows. First we assume that the rod is mass less and that the cart mass and the point mass at the upper end of the inverted pendulum are denoted as M and m, respectively. There is an externally x-directed force on the cart u(t), and a gravity force acts on the point mass at all times, where x(t) represents the cart position and θ(t) is the tilt angle referenced to the vertically upward direction. Here we use Lagrangian model for analysis. In this we have to find the kinetic energy and potential energy. The Lagrangian will be the difference of the kinetic energy and potential energy. L=T −V (4) where, T = Kinetic Energy(KE), V = Potential Energy(PE)

(3)

0

The Caputo and Riemann-Liouville formulation coincide when the initial conditions are zero. Properties of Fractional Derivatives (Igor Podlubny (1999)):- and, Linearity:Dp (λf (t) + µg(t)) = λDp f (t) + µDp g(t) The most important property of the RL fractional derivative is that for p > 0 and t > 0, Dtp (Dt−p f (t)) = f (t) which means that RL fractional differentiation operator is left inverse to the RL fractional integration operator of the same order p. Laplace Transformation of Fractional Derivatives:L(Dt−p f (t); s) = s−p F (s)

Ttotal =

Vtotal = Vcart + Vpendulum

(5)

Vtotal = mgl cos θ,

(6)

Ttotal = Tcart + Tpendulum

(7)

1 1 ˙ cos θ) M x˙ 2 + m(x˙ 2 + l2 θ˙2 + 2x˙ θl 2 2

(8)

Here, M = mass of the cart( (Kg)). m= mass of pendulum ((Kg)), g= acceleration due to gravity ((m/s2 )), h= height of the pendulum (M eter) =l cos θ, x = displacement of cart (M eter) The Lagrangian will be;

L = Ttotal − Vtotal

L=

(9)

1 1 ˙ cos θ)−mgl cos θ (10) (M +m)x˙ 2 + m(l2 θ˙2 +2x˙ θl 2 2

Put equation (10) into Euler-Lagrange equation below, d ∂L ∂L ( )− =u dt ∂ x˙ ∂x

(11)

and ∂L d ∂L ( )− =0 dt ∂ θ˙ ∂θ

(12)

We get,

(M + m)¨ x + ml cos θθ¨ − ml sin θθ˙2 = u

(13)

lθ¨ + x ¨ cos θ − g sin θ = 0

(14)

After separating the values of x ¨ and θ¨ we get; (u cos θ + ml sin θ cos θθ˙2 − (M + m)s sin θ) θ¨ = ml cos2 θ − (M + m)l (u + ml sin θθ˙2 ) − mg sin θ cos θ x ¨= M + m − m cos2 θ

(19)

After solving the above we get the linearised matrix as,     0 1 0 0 0    1  −mg 0 0 0   d M  δz +  M  δu (20) (δz) =     0  dt 0 1 0 0   0 0

(M +m)g Ml

(15) y= (16)

(17)

Let, z1 = x z2 = x˙ = z˙1 z3 = θ z4 = θ˙ = z˙3 Resulting nonlinear state space representation is expressed as follows,   z  z2 1   (u+ml sin θ θ˙ 2 )−mg sin θ cos θ    d  M +m−m cos2 θ z2     =   z3 dt z3     2 (u cos θ+ml sin θ cos θ θ˙ −(M +m)s sin θ) z4 2

d (δz) = J=z (z0 , u0 )δz + J=u (z0 , u0 )δu dt

0

−1 Ml

And the output matrix as,

We cast (15) and (16) into standard non-linear state-space representation as:d (z) = f (z, u, t) dt

Fig. 2. POAC system’s block diagram.

(18)

ml cos θ−(M +m)l

We now linearize the above state-space by taking Jacobian at (z0 , u0 ) = (0, 0) [J. R. White (1997)] The linearized form of the system becomes,

  x !  1 0 0 0 x˙     0 0 1 0  θ  θ˙

(21)

3. PID AND FOPID CONTROLLER DESIGN Fig.(2) shows the block diagram of the whole controlled system (POAC) and accordingly we design the controller. For the purpose of simulation, we consider following system parameters, m = 0.2Kg, M = 0.5Kg, l = 1M eter, g = 9.8m/s2 . Integer order PID controller is of the form as below, ki kpid = kp + + kd s (22) s Fractional order PID controller is of the form as below, ki kf opid = kp + α + kd sβ (23) s where, 0 < α < 1, and 0 < β < 1 Clearly, selecting α = β = 1, a classical PID controller can be recovered. The selections of α = 1,β = 0, α = 0,β = 1, respectively corresponds conventional PI and PD controllers. All these classical types of PID controllers are

Fig. 3. Fractional order controlled response.

Fig. 4. Integer order controlled response.

the special cases of the fractional P I α Dβ controller. It can be expected that the controller P I α Dβ may enhance the systems control performance[J.Samardzic, M.P.Lazarevic, B. Cvetkovic (2011)]. Since this kind of controller has five parameters to tune (Kp , Kd , Ki , α, β), up to five design specifications for the controlled system can be met, that is, two more than in the case of a conventional PID controller, where α = 1 and β = 1. It is essential to study which specifications are more interesting as far as performance and robustness are concerned, since it is the aim to obtain a controlled system robust to uncertainties of the plant model, load disturbances, and high-frequency noise. All these constraints will be taken into account in the tuning technique in order to take advantage of the introduction of the fractional orders (C. A. Monje, YangQuan Chen, B. M. Vinagre, Dingy Xue, Vicente Feliu. (2010)). We propose FOPID controller for the state-space model as well as transfer function model of the system under consideration. We first propose optimal conventional PID controller and then exploit the freedom in FOPID to improve the performance. We first consider the state-space model obtained in equation (20) and (21), to select the parameters using integer PID tuning. For the purpose of comparison we keep same tuning parameters in FOPID and use freedom in selecting α and β. The optimal values of the parameters are as follows:

Fig. 5. Cart’s position control comparison.

Table 1. Optimal values of parameters (Pendulum’s angle control) Controller FOPID PID

α 0.95 1

β 0.9 1

kp -55 -55

kd -8 -8

ki -25 -25

Table 2. Optimal values of parameters (Cart’s position control) Controller FOPID PID

α .009 1

β .8 1

kp -1.7416 -1.7416

kd -3.192 -3.192

ki -0.0128 -0.0128

Fig. 6. Pendulum’s angle control comparison. Fig.(3) shows the Fractional order PID response and Fig.(4) shows the Integer Order PID response for the system under consideration with the objective of controlling the cart position and stabilizing pendulum angle. From Fig.(5) we may conclude that, there is some oscillation in the Integer Order PID controller for cart’s position and that can be compensated by using the Fractional Order PID controller. From Fig.(6), we observe that there is more oscillation and a larger overshoot in Integer Order PID controller for

Fig. 7. Cart’s position control comparison under disturbance.

Fig. 9. Fractional and Integer order controlled response. Compensation Goals:The desired transient response for the system has the following characteristics: 1) Transient (settling) time of 10 second. 2) Overshoot should be < 30 percentage. The desired steady-state response for the system has following characteristics: 1) Steady-state error must be zero.

Fig. 8. Pendulum’s angle control comparison under disturbance. pendulum’s angle as compared to Fractional Order PID controller. To investigate the robustness of the proposed controller, we apply a unit step disturbance to the cart and compare the results of Fractional Order PID controller to the Integer Order PID controller. From Fig.(7) and Fig.(8), we observe the difference in the performance of fractional and integer order controller in the presence of disturbances. Simulations further show that the fractional order PID controller performs better compared to integer order PID controller. 3.1 Systematic approach to find parameters:We consider pendulum’s vertical upward position as equilibrium and linearize equation (13) and (14) around this equilibrium. After linearizing we get, (M + m)¨ x + mlθ¨ = u (24) ml2 θ¨ + ml¨ x − mglθ = 0

(25)

We obtain the transfer function of the system with initial conditions assumed to be zero, as follows: θ(s) 1 = U (s) (M + m)g − M ls2

(26)

Compensation Design:- The compensator designing has been done with MATLAB SISO DESIGN TOOL. From the SISO design tool we obtain the parameters (kp , kd and ki ) of the PID controller and the same values of parameter will be used to design the FOPID controller as for comparison purpose. Independent selection of FOPID may give even better performance. The transfer function of the system (26) takes following form after substituting system parameters: −1 G(s) = (27) 0.5s2 − 6.86 and the PID compensator obtained by SISO tool is:6.4 C(s)P ID = −33.28 − − 6.4s (28) s Which gives kp = −33.28,kd = −6.4 and ki = −6.4. We consider the same values of kp , kd and ki for FOPID and use additional freedom of tuning α and β to improve the performance of FOPID. Although, the systematic procedure for obtaining the values of α and β is still under consideration[YangQuan Chen (2010)], we observe that for the model under consideration, the choice of α = 0.8 and β = 0.4 gives the best results. 6.4 C(s)F OP ID = −33.28 − α − 6.4sβ (29) s where the α and β are the fractional power of integrator and differentiator respectively. The comparison results for optimum PID and FOPID are shown in Fig.(9) It is observed from Fig.(9) that the FOPID performs better than the integer order PID. This is not surprising since FOPID has two more extra parameters in optimal search. 4. CONCLUSION In this paper, we successfully stabilized the POAC system using Fractional Order (FO) PID controller. The objective was set to control the angle of the pendulum and the position of the cart. Also we have compared the results of FOPID controller and Integer Order PID controller. It was observed through

the simulation results that the FOPID controller is better for stabilizing the inverted pendulum on a cart system. Future work is aimed at experimentally validating the results obtained by the simulation and arriving at a systematic procedure to obtain the FOPID parameters.

REFERENCES [1] Ahmad Nor Kasruddin Bin Nasir ”Modelling and Controller design for an Inverted Pendulum System”. ”Department of Electrical,Universiti Teknologi Malaysia”,114,(2006-07). [2] Duarte Pedro Mata de Oliveira Valrio. User and programmer manual. ”Ninteger v. 2.3 Fractional control toolbox for MatLab”,(2005/08). [3] Dr. J. R. White, UMass-Lowell . Lecture Notes ”Introduction to the Design and Simulation of Controlled Systems” 24.509,(Spring 1997). [4] Manue F. Silva, J. A. Tenreiro Machado. Fractional Order P Dα Joint Control of Legged Robots ”Journal of Vibration and Control,12(12)”,1483-1501,(2006). [5] Manue F. Silva, J. A. Tenreiro Machado and A. M. Lopes. Fractional Order Control of a Hexapod Robot ”Mendeley”,vol-38,issue:1-4,417-433, (2003-04). [6] Mahmut Reyhanoglu,Arjan van der Schaft,N. Harris McClamroch and Ilya Kolmanovsky. Dynamics and Control of a Class of Underactuated Mechanical Systems. ”IEEE Transactions on automatic control”,vol44,no.9,1683-1671, (1999). [7] C. Chevallereau, J.W. Grizzle and C.H. Moog. Nonlinear Control of Mechanical Systems with one Degree of Underactuation. ”Robotics and automation”,vol3,2222-2228, (2004). [8] Subhransu Padhee, Abhinav Gautam, Yaduvir Singh, and Gagandeep Kaur. A Novel Evolutionary Tuning Method for Fractional Order PID Controller. ”International journal of soft-computing and engineering”,vol1,1-9,(2011). [9] Duarte Valrio, Jos S da Costa. Non-Integer order control of a flexible robot. ”Technical university of Lisbon”,1-6, (2004). [10] Hyo-Sung Ahn, Varsha Bhambhani and YangQuan Chen. Fractional-order integral and derivative controller design for temperature profile control. ”Control And Decision Conference,CCDC”,4766-4771, (2008). [11] J. L. Adams, T. T. Hartley and C. F. Lorenzo. Fractional-order system identification using complex order-distributions. ”IFAC”,vol-2,prt-1,(2006). [12] Igor Podlubny. Fractional Differential Equations (1999). [13] C. A. Monje, YangQuan Chen,B. M. Vinagre, Dingy Xue, Vicente Feliu. ”Fractional-order Systems and Controls Fundamentals and Applications”. (2010). [14] J.Samardzic, M.P.Lazarevic, B. Cvetkovic. Optimal conventional and fractional PID control algorithm for a robotic system with three degrees of freedom driven by DC motors. Intelligent systems and informatics (SICY)”,297-301, (2011). [15] YangQuan Chen. Ubiquitous Fractional Order Controls. Second IFAC Workshop on Fractional Differentiation and its Applications, 2006 , Volume-2, Part-1. [16] Faruk Kazi,Ravi N. Banavar,Philippe Mullhaupt,Dominique. Stabilization of a 2D-SpiderCrane

Mechanism using Damping Assignment Passivity-based Control, ”IFAC world congress”,3155-3160, (2008). [17] Ioannis Sarras, Faruk Kazi, Romeo Ortega and Ravi Banavar.. Total Energy-Shaping IDAPBC Control of the 2D-SpiderCrane. ”Decision and Control (CDC)”,1122- 1127, (2010). [18] YangQuan Chen, B. M. Vinagre, and Igor Podlubny. On fractional order disturbance observers. In Proc. of The First Symposium on Fractional Derivatives and Their Applications at The 19th Biennial Conference on Mechanical Vibration and Noise, the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (ASME DETC2003), pages 18, DETC2003/VIB–48371, Chicago, Illinois, 2003. [19] YangQuan Chen, Kevin L. Moore, Blas M. Vinagre, and Igor Podlubny. Robust PID controller autotuning with a phase shaper. In Proceedings of The First IFAC Symposium on Fractional Differentiation and its Applications (FDA04), Bordeaux, France, 2004a. [20] YangQuan Chen, Blas M. Vinagre, and Igor Podlubny. Continued fraction expansion approaches to discretizing fractional order derivatives - an expository review. Nonlinear Dynamics (Kluwer), 9(1): to appear, 2004b. [21] YangQuan Chen, Dingy Xue, and Huifang Dou. Fractional calculus and biomimetic control. In Proc. of the First IEEE Int. Conf. on Robotics and Biomimetics (RoBio04), pages robio2004- 347, Shengyang, China, August 2004c. IEEE. [22] Jinsong Liang, YangQuan Chen, Blas M. Vinagre, and Igor Podlubny. Boundary stabilization of a fractional wave equation via a fractional order boundary controller. In Proceedings of The First IFAC Symposium on Fractional Differentiation and its Applications, Bordeaux, France, July 19-20, 2004b. [23] Jinsong Liang, YangQuan Chen, Blas M. Vinagre, and Igor Podlubny. Fractional order boundary stabilization of a time-fractional wave equation. a chapter in Fractional Derivatives an Their Applications. Editors: A. Le Mehaut, J. A. Tenreiro Machado, J. C. Trigeassou and J. Sabatier. UBooks, Augsburg, Germany, 2005. [24] Igor Podlubny. Fractional-order systems and P I α Dβ controllers. IEEE Trans. Automatic Control, 44(1): 208214, 1999.