INPUT-STATE LINEARIZATION OF A ROTARY INVERTED PENDULUM

Transcription

1 0 Asian Journal of Control Vol 6 No pp 0-5 March 004 Brief Paper INPU-SAE LINEARIZAION OF A ROARY INVERED PENDULUM Chih-Keng Chen Chih-Jer Lin and Liang-Chun Yao ABSRAC he aim of this paper is to design a nonlinear controller for the rotary inverted pendulum system using the input-state linearization method he system is linearized and the conditions necessary for the system to be linearizable are discussed he range of the equilibriums of the system is also investigated Further after the system is linearized the linear servo controllers are designed based on the pole-placement scheme to control the output tracking problem he performance of the controller is studied with different system parameters he computer simulations demonstrate that the controller can effectively track the reference inputs KeyWords: Input-state linearization nonlinear control rotary inverted pendulum pole-placement method I INRODUCION he rotary inverted pendulum is a widely investigated nonlinear system due to its static instability his paper deals with a rotary inverted pendulum system (see Fig ) which is composed of the following components: a rotating disk which is driven by a motor with a rod mounted on its rim he rod moves as an inverted pendulum in a plane perpendicular to the rotating disk he system discussed in this paper is not the same as the normal inverted pendulum or Furuta s pendulum In contrast to the later two systems in which the equilibria are at two points (with the pendulum upright or down vertically) the equilibria of this system are important and will be investigated in the paper he system is used as a simplified model for the control of rider-motorcycle systems in circular motion on paths of different radii he results of this research can be used to study how the riding speeds of the motorcycle and the leaning angles of the rider affect the system dynamics and its motion Wu and Liu [] used genetic algorithm and auto-tuning to improve the performance of a fuzzy controller for their system Yurkovich and Widjaja [] dealt with the control problem using two strategies For swing-up control a fuzzy supervisory mechanism was used For balancing Manuscript received August 8 00; revised March 5 00; accepted April 8 00 he authors are with Department of Mechanical and Automation Engineering Da-Yeh University Changhua aiwan 5505 ROC Inverted Pendulum Mass m = 004kg Disk Mass M = 6kg θ m Motor l = 0m r = 0m Fig A schematic representation of the rotary inverted pendulum control a direct fuzzy controller based on the LQG linear control was utilized hese methods do not directly use the nonlinear mathematical model in the controller design Feedback linearization methods can be viewed as ways of algebraically transforming a nonlinear system dynamic fully or partially into a simple linear one In the standard approach to exact input-state linearization one uses coordinate transformation and static state feedback such that the closed-loop system in the defined region θ b

2 CK Chen et al: Input-State Linearization of a Rotary Inverted Pendulum takes a linear canonical form After the system s linearization form is obtained the linear control design scheme is empoloyed to achieve stabilization or tracking [57] Many applications can be found in the literature [-5] he problem dealt with in this paper is that of finding a control law to be applied to the motor s torque such that the inverted pendulum motion stays at a specific point and tracks the desired trajectory he system is linearized using the input-state linearization method [5] and the properties of the linearized system are discussed in this paper Further the linear control law is designed using pole-placement scheme to make the output states track the desired trajectory he integral control law is used in the design to eliminate steady-state errors caused by sys- tem uncertainties he performance of the controller is studied using different system parameters and the simulation results verify the capability of the controller II DYNAMIC MODEL In this paper we investigate the motion control of a rotary inverted pendulum as shown in Fig by means of input-state linearization Denote by θ m the angular velocity of the rotation disk which is controlled by the motor torque and by θ b the angle between the rod and the vertical axis which is the output state of the system Denote by r the radius of the disk by M the mass of the disk by l the length of the rod and by m the mass of the rod hen the Lagrange function L is defined as L = V () where is the kinetic energy and V is the potential energy From Lagrange s equation we can have the equations of motion as d L L d L L = u = 0 dt θ m θm dt θ b θb where u is the control torque applied on the rotating disk By substituting the Lagrange function L in () into the equations in () the dynamic equations of the system can be obtained as follows: ml cosθθ b mθ b( l sinθb) + 6u ( + 6 ) + sinθb( sinθb 6 ) r M m ml l r () θ = () m cos θθ b m (lsinθb r) gsinθb θ + b = (4) l Now we can go further to study the control problem for the pendulum angle θ b he system dynamics in Equations () and (4) are nonlinear; thus the linearization procedure can be utilized to design the controller III INPU-SAE LINEARIZAION he standard form of the nonlinear system can be written as x= f( x) + g( x ) u (5) where f(x) and g(x) are smooth vector fields on R n If the state vector is defined as x = [ x x x] = [ θ m θb θ b] then Equations () and (4) can be rewritten in the standard form as in (5) with the vector fields defined as follows: ml cos xxx( l sin x) D fx ( ) = x cos xx (lsin x r) + gsin x 6 gx ( ) = 0 0 D where D = r ( M + m) + mlsin x( lsin x r) he definition of input-state linearization is given in the following One can apply the standard procedure of the method to linearize the nonlinear system (5) Definition [5] A nonlinear system in the form of (5) is said to be input-state linearizable if there exists a region Ω in R n n a diffeomorphism φ : Ω R and a nonlinear feedback control law l (6) u = α( x) + β ( x ) ν (7) such that the new state variables z = φ ( x) and the new input ν satisfy a linear time-invariant relation: z = Az+ b ν (8) he new state vector z is the linearized state and (7) is called the linearizing control law By the above definition the fundamental results of feedback linearization can be used to linearize the system Lemma [56] he nonlinear system (5) with f(x) and g(x) being smooth vector fields is input-state linearizable if and only if there exists a region Ω such that the following conditions hold: n the vector fields { g ad fg ad f g } are linearly independent in Ω n the set { g ad fg ad f g } is involutive in the region Ω

3 Asian Journal of Control Vol 6 No March 004 heorem he nonlinear system (6) can be linearized as the linear equation (8) in the following region: Ω x R 0 ( ) and sin( x ) k Z l = x x k + Proof We will show that the system can satisfy the conditions in Lemma First for the nonlinear system (6) the set of vector fields can be described as {g ad f g ad f g} and the determinant of the three vector fields can be obtained as follows: det([ g ad fg ad f g ]) = (9) 6x cos x ( lsin x ) Dl (0) If det([ g ad fg ad f g ]) 0 then the vector fields in the set { g ad fg ad f g } are linear independent his leads to the following conditions: x 0 cos x 0 and ( lsin x ) 0 Secondly one can show that the set of vectorfields { g ad f g } are involutive By substituting f and g into Equation (6) the vectorfields ad f g can be obtained as follows: ad f g 6xcos x(lsinx r) 0 0 = Dl Furthermore the Lie bracket of g and ad f g can be computed as follows: [ g ad f g ] It is obvious that ( ) 6cos x lsin x = 0 0 D l [ g ad fg] = αg+ α ad fg where α α are scalar functions defined as α = 0 α = 6x herefore the set of vectorfields { g } D ad f g is involutive in Ω he above satisfaction the two conditions in Lemma proves that the system is linearizable using input-state linearization method Since the nonlinear system (5) satisfies the conditions in Lemma the steps needed to obtain the equivalent linear system by means of input-state linearization are as follows: Lemma [7] he nonlinear system (5) that satisfies the conditions in Lemma can be transferred to the linear system (8) by means of the following state transformations: ransfer the state variables x to z with the first state z satisfying i z g = z adf g = 0 i = 0 n zad 0 n f g () he new state vector is ( ) [ n zx = z Lfz Lf z] and the control command is u = α( x) + β ( x ) ν where Lf z ( ) α x = and β ( x ) = LL g f z LL g f z Lemma can be applied directly to linearize the system as shown in the following theorem heorem Choose the new state vector z and the control law u as follows: [ ] z = x x x u = α( x) + β ( x ) ν; () then system (6) can be linearized to obtain the linear system (8) in the region Ω as defined in (9) with and E α( x ) = x cos x (lsin x r) ld β ( x ) = x cos x (lsin x r) E = lx x cos x (6Mr ml + 4ml cosx + 6mr + 8mrl sin x ) + 9rx x sin x ( Mr + ml + mr ) + gcos x x D lx x (5mr + Mr + ml ) Proof From the computations of the vector fields we have D = 0 0 P 0 P 0 [ g ad fg ad f g ] 6x cos x ( lsin x ) where P Dl tion () in Lemma the first new state z of system (6) must satisfy the following equations: = o satisfy Equa- z z z = 0 = 0 0 x x x

4 CK Chen et al: Input-State Linearization of a Rotary Inverted Pendulum he simplest solution to the above equations is z = x; then the conditions x = 0 z g = z ad g = 0 zad f g = f 6xcos x(lsinx r) Dl x = x = can be satisfied From the above if the nonlinear system (6) stays in the region Ω defined in Equation (9) then zad f g 0 and the other two states z z can be obtained from z : z = zf = x cos xx (lsin x r) + gsin x z = zf = x = l he control law u given in Lemma is u = α(x) + β(x)ν Lf z where ( ) α x = and β ( x ) = By LL g f z LL g f z letting z = x and computing the above two equations one can obtain the result of Equation () As a result of the above state transformations we can transform the system dynamics into the linear form as shown in (8) with the parameters x = sin (/l) x = x = + sin (/l) x = (a) l x = 0 x = A = b = () and with the state transformations between z and x as follows: (gsin z lz) θ m = x =± cos z lsin z θ b = x = z θ = x = z b ( ) he new state variables are as follows: z = [z z z ] = [ θb θ b θ b] hese are the angular position velocity and acceleration of the pendulum Remark For the nonlinear control system (5) the equilibrium point x * is defined by f(x * ) = 0 without the control effort ie u = 0 For system (6) the equilibrium point can be obtained as gtanx * x = ± x 0 (4) lsinx Since the angular velocity of the disk x = θ m cannot be imaginary the condition for the existence of the equilibrium is x = (b) < l Fig he regions of the equilibrium points gtanx 0 lsinx > (5) Condition (5) leads to two situations: () tan x 0 and lsinx > 0 ; () tan x 0 and l sin x < 0 In each case the equilibrium points are located in the following domains: () When l the equilibrium points are in the region shown in Fig (a) () When < l the equilibrium points are in the region shown in Fig (b) Referring to the configuration of the pendulum the origin of angle x is defined to correspond to the situation where the rod stays vertically upwards as shown in Fig he parameters r and l are the disk radius and the length of the pendulum respectively he condition < l in case () can be interpreted as the case where the relatively long rod is used as the pendulum and the centrifugal force lifts the rod to counteract gravity in the region < x < + sin ( ) In most practical l

5 4 Asian Journal of Control Vol 6 No March 004 control problems the angles chosen as the reference inputs are within the range 0 < x < hen one must keep in mind that for the long rod case ( < l) the range sin ( ) < x < is not the equilibria and l the pendulum cannot be controlled such that it stays in the region Remark In the linearizable region in Equation (9) the point x = (k+ ) or sin( x ) = is at the bound- l ary of the region shown in Fig Furthermore x = 0 means that the disk stops rotating hese points lead to the singularity condition for input/state feedback linearization Since state variable transformation is involved in the process of input-state linearization the choice of new states is crucial Different choices of states may lead to different complexity of the nonlinear controller In this paper we obtain a set of simple state variables that can effectively transform the nonlinear system into a simple linear one without loss of physical meaning IV CONROLLER DESIGN AND SIMULAIONS he nonlinear system dynamics were transformed into an equivalent linear system in the normal form (8) In this study the linear controller design schemes were used to control the pendulum he integral-type servo controller was used to improve the performance of tracking control of the pendulum angle 4 Linear controller design A block diagram of the controller is shown in Fig Here the integral control is not considered (ie k I = 0) For the linearized system the linear state feedback control law can be designed as follows: r ν = Kz +kr (6) + _ e + v k z = Az + bv + k I s k k z y = z y = Cz Fig A block diagram of the linear servo control system he feedback gain K can be designed using the pole-placement method o minimize the IAE criterion for the step input one can choose dominant poles with a damping ratio of ξ = 0707 and a natural frequency of ω n = 0 he three system poles can consequently be obtained as ( 707±707j 707) he system s nominal parameters are described in able and the feedback gain K can be computed as follows: where the feedback gains are K = [ k k k] [ ] K = 0e (7) However the performance for the tracking problem degrades due to the parameter uncertainties in the system One could improve the steady-state error by adding the k I integral controller as shown in Fig If the same s feedback gain K as in (7) is employed and the integral gain is k I = 0000 then the closed-loop poles of the system are at ( 407±549j ) In addition saturation was also considered in the simulations For practical implementation of the system the motor s driving torque was assumed to be restricted within the range ±7 (N-M); thus the saturator was added to the simulations 4 Simulation results he system parameters are listed in able o investigate the effect of the modeling uncertainties on the controller performance two sets of data were used he nominal parameters were used in the nonlinear controller design and the actual parameters were used in the simulation of the plant he reference inputs were θ b = 5(deg) 0 t < (sec) θ b = 5 (deg) t 4 (sec) and initial system conditions of θ b (0) = 0 (deg) θ b (0) = 0 (deg/sec) and θ m (0) = 0 (deg/sec) were used to study the step responses In Fig 4 the results of three simulation cases are plotted For Cases and as shown in the figure a control saturation limit of u 7 was able he system nominal parameters and actual parameters System Actual Nominal parameters parameters parameters M 6kg 6kg m 00kg 004kg l 05m 0m r 05m 0m

6 CK Chen et al: Input-State Linearization of a Rotary Inverted Pendulum Refrence input Case : ki = 0 with saturation Case : ki = 0000 without saturation Case : ki = 0000 with saturation Refrence input Case 4: ki = 0 Case 5: ki = θ b (deg) time (sec) Fig 4 Step responses of the inverted pendulum angle applied and the angular speed of the disk cannot be built up in time; thus the pendulum first fell and then swung up due to the centrifugal force Without control saturation (Case ) the pendulum could be controlled directly from an initial position ranging from 0 to 5 However the maximum control torque reached to 9 N-M which is far beyond the constraint of 7 N-M As mentioned above in the saturated case the disk speed could not be built in time to counteract the force of gravity; thus the controller produced higher speed to lift the pendulum For these cases the disk s angular speed converged to the equilibrium point as shown in (4) In Case the integral controller was not used (ie k I = 0) and it can be seen that a steady-state error existed in the response due to system uncertainties his error could be reduced by adding the integral controller as shown in the Cases and he simulation results show the capability of the integral controller in eliminating the tracking error under parameter uncertainties In addition with the same initial conditions we further studied the tracking problem he desired angular trajectory of the pendulum was given by the sinusoidal function 0 + 0sin(t) (deg) he time responses of the pendulum to the sinusoidal input are plotted in Fig 5 As shown by these plots modeling uncertainties still existed in the plant Both of the responses (Cases 4 and 5) to the time-variant reference input showd steady-state errors due to system uncertainties However the tracking error in the Case 5 could be effectively reduced due to the effect of the integral controller V CONCLUSIONS In this research nonlinear control design for a rotary inverted pendulum system has been studied using the input-state linearization method he system has been linearized to obtain a linear system of the normal θ b (deg) time (sec) Fig 5 ime responses of the inverted pendulum angle to the sinusoidal input form and a linear controller has been used to achieve tracking control of the pendulum s angular position Simulations were conducted to verify the control results When the input-state linearization method was applied to the system the region of equilibrium points of the system was affected by the geometric parameters his shows that systems with different parameter ranges ( l or < l) will have different equilibrium regions he parameters uncertainties resulted in a tracking error By adding the integral controller to the system it was possible to reduce the steady state error In future research the robust controller will be incorporated to improve the tracking accuracy when time-varying parameters or disturbances exist REFERENCES Wu J C and S Liu Fuzzy Control of Rider-Motorcycle System Using Genetic Algorithm and Auto-uning Mechatronics Vol 5 No 4 pp (995) Yurkovich S and M Widjaja Fuzzy Control Synthesis for An Inverted Pendulum System Contr Eng Parct Vol 4 No 4 pp (996) Fattah H A Input-output Linearization of Induction Motors with Magnetic Saturation in Proc Amer Contr Conf pp (000) 4 Jakubczyk B and W Respondek On Linearization of Control Systems Bull Acad Polonaise Sci Ser Sci Math 8 pp 57-5 (980) 5 Slotine J E and W P Li Applied Nonlinear Control Prentice-Hall NJ (99) 6 Su R On the Linear Equivalents of Nonlinear Systems Syst Contr Lett Vol No pp 48-5 (98) 7 Isidori A Nonlinear Control Systems d Ed Springer New York (995)