A DSP Microprocessor Hybrid Control of an Inverted Pendulum

2009 IEEE International Conference on Control and Automation Christchurch, New Zealand, December 9-11, 2009

FrPT3.1

A DSP Microprocessor Hybrid Control of an Inverted Pendulum Ho-Lung Li, Bor-Chin Chang, Chirag Jagadish, and Harry G. Kwatny Abstract— A hybrid control system implemented in the DSP microprocessor TMS320F2812 is designed to globally stabilize the system for all possible initial conditions. A set of design guidelines for swing-up control and supervisory coordination has been employed in the simulation and real-time experiment. The simulation results and the real-time performance graphs are also presented.

I. INTRODUCTION HE inverted pendulum control system is one of the most well known benchmark control problems that have been extensively studied for decades, for details please see [1, 2] and the references therein. The problem is so intriguingly interesting because it is inherently unstable, nonlinear, under-actuated, and almost all available control design approaches have been employed to solve the problem. These approaches include LQG approach [3], nonlinear H ∞ control [4], feedback linearization [5, 6], hybrid control [2, 7], energy-based approach [1, 2, 8, 9], fuzzy logic [10, 11], bang-bang control [12], sliding mode control [13], and neural networks [14], etc. In this paper, we will focus on the DSP microprocessor implementation of a hybrid control system that would globally stabilize the system for all initial conditions and possible non-persistent disturbances. The mechanical cart-pendulum hardware used is the Quanser High Fidelity Linear Cart [15], which includes precision optical encoders that can be used together with the Texas Instruments DSP microprocessor TMS320F2812 [16] to measure the angular displacement and velocity of the pendulum and the translational displacement and velocity of the cart. The cart is driven by a 3-phase 400W brushless DC motor that is capable of delivering 0.7 N-m torque continuously. A brushless PWM servo amplifier [17] is employed to serve as the interface between the PWM control signal from the DSP microprocessor and the brushless DC motor. The hybrid control system is implemented in the same DSP microprocessor chip. The program is written in C and the development software is the Code Composer Studio IDE of Texas Instruments. The structure of the hybrid control system to be used is

T

Manuscript received April 15, 2009. This work was supported in part by the U.S. Army Research Laboratory under contract DAAD17-03-P-0422, and in part by NASA under contract NNX09CE93P. H.-L. Li is with the Mechanical Engineering Department, Drexel University, Philadelphia, PA 19104 USA ([email protected]). B.-C. Chang is with the Mechanical Engineering Department, Drexel University, Philadelphia, PA 19104 USA (Corresponding author, 215-895-1790, [email protected]). C. Jagadish was with is with the Mechanical Engineering Department, Drexel University, Philadelphia, PA 19104 USA ([email protected]). H. G. Kwatny is with the Mechanical Engineering Department, Drexel University, Philadelphia, PA 19104 USA ([email protected])

978-1-4244-4707-7/09/$25.00 ©2009 IEEE

similar to those in most of the swing-up inverted pendulum articles in the literature [1, 2]. It consists of a linear stabilizing controller, a swing-up control unit, and a supervisory coordinator. The linear stabilizing controller is designed based on a linearized model of the inverted pendulum around the saddle equilibrium point where the pendulum is at the upward position and the cart is rest at the middle of the rail. The linear controller is supposed to be able to stabilize the system at the equilibrium as long as the system is inside the neighborhood of the equilibrium where the system dynamics is approximately linear. This neighborhood is called the region of attraction for the linear stabilizing controller. When the system is outside the region of attraction, the supervisory coordinator will temporarily disengage the linear controller from the system and may request the swing-up control unit to step in to swing the pendulum back to the region of attraction. Many swing-up strategies have been proposed [8, 12]; the basic idea is to move the cart strategically to pump an adequate amount of energy to the pendulum so that it can swing up to the upward position with negligible kinetic energy at the top. The swing-up strategy we will use is to control the motion of the cart so that the cart movement synchronizes with the pendulum motion in a way that the swing-up motion can be optimized and the cart position is always inside the rail limit. The supervisory coordinator will watch over the overall condition and determine the operating mode for system. Most of the time, it will be a switching between the linear controller and the swing-up control unit. However, there may be a period of time the system would perform better if both of the two subsystems remain inactive. The rest of this paper is organized as follows. In section II, the nonlinear dynamics equations and state-space model of the inverted pendulum system are derived. A phase portrait of the nonlinear model shows that there are two equilibrium points – one associated with the upward pendulum position which is unstable and another with the stable downward pendulum. A linearized state-space model will also be derived to represent the system at the neighborhood around the unstable equilibrium. In section III, the linearized model will be used to design a linear stabilizing controller. The strategies for the swing-up control unit and the supervisory coordinator will also be developed in this section. In section IV, simulations of the closed-loop system that consists of the nonlinear model of section I and the hybrid control system designed in section III will be performed to verify the design concept. Section V will present the real-time performance of the DSP microprocessor hybrid control system on the Quanser inverted pendulum mechanical system. Concluding remarks are given in Section VI.

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A. Nonlinear state-space model of the inverted pendulum A schematic diagram of a typical linear-rail inverted pendulum system is shown in Fig.2.1. The cart has mass M kilo-grams and is driven by a force u Newtons, which is generated by a brushless DC motor inside the cart. The displacement of the cart is represented by s meters measuring from the center of the rail to the right. A positive (negative, resp.) s means the cart is at the right (left, resp.) side of the rail. The direction of the force is defined accordingly so that a positive force u would move the cart forward to the right, and a negative u would reverse the cart to go left. The friction coefficient of the translational motion is assumed as a constant Bs. 2A mg

θ

s Mg

u

Fig. 2.1. An inverted pendulum system

The stick with mass m kilo-grams and length 2A meters is hinged to the cart with a pivot so that the stick can rotate freely and make whole 360-degree swing in either clockwise (c.w.) or counter clockwise (c.c.w.) direction. The friction coefficient of the rotational motion is assumed as a constant Bθ. The angular displacement is represented by θ radians measuring from the upward reference in the c.c.w. direction. A positive (negative, resp.) θ means the stick is tilted to the left (right, resp.) side of the pivot. Based on Newton’s law of motion, the governing dynamic equations for the system can be found as follows. (4 / 3)mA 2θ + Bθ θ − mg A sin θ − mA cos θ  s=0 (2.1a) 2 ( M + m) s + B s + mA sin θθ − mA cos θθ = u (2.1b) s

With M=1.79kg, m=0.104kg, A =0.3048m, g=9.8m/s2, Bs=0.25Ns/m, and Bθ =0.02Ns, Eqs. (2.1a-b) become 0.01288θ + 0.02θ − 0.3107 sin θ − 0.0317 cos θ  s = 0 (2.2a) 1.894 s + 0.25s + 0.0317 sin θθ 2 − 0.0317 cos θθ = u (2.2b) Define state variables x1=θ, x2= θ , x3=s, and x4= s , then Eqs. (2.2a-b) can be converted into the following nonlinear state equation. (2.3) xi (t ) = f i ( x) + gi ( x)u , i = 1, 2,3, 4 where f1 ( x) = x2 , f3 ( x) = x4 f 2 ( x) = (1/ Δ )(5885sin x1 − 378.8 x2 − 10sin x1 cos x1 x22 − 79.25 x4 cos x1 )

f 4 ( x) = (1/ Δ)(−4.083sin x1 x22 − 32.2 x4 +98.5sin x1 cos x1 − 6.34 x2 cos x1 ) g1 ( x) = g3 ( x) = 0 , g 2 ( x) = (1/ Δ )(317 cos x1 ) g 4 ( x) = (1/ Δ )(128.8) and Δ = 244 − 10 cos 2 x1 . Fig.2.2 shows the phase portrait of the nonlinear system

based on Eq. (2-3). It can be seen that, the upward position (2kπ, 0) is unstable and the downward position of the pendulum ((2k+1)π, 0) is stable. 10

5 x2, rad/s

II. MATHEMATICAL MODELS

0

-5

-10 -6

-4

-2

0 x1, rad

2

4

6

Fig. 2.2. Phase portrait of the uncompensated nonlinear pendulum.

B. Linearized state-space model of the inverted pendulum system around the unstable equilibrium The above nonlinear state-space model of the inverted pendulum will be employed in the simulation for all operating conditions including swing-up and the stabilization around the originally unstable equilibrium. The following linearized state-space model will be used in the design of a linear stabilizing controller that would stabilize the pendulum at the upward position as long as the pendulum is in the linear regime. Applying the Jacobian linearization approach [3, 18] to the nonlinear equations Eq. (2-3), a linearized state-space model can be easily obtained as follows. G p ( s ) : x (t ) = Ap x(t ) + B p u (t ) (2-4)

where 1 ª 0 « 25.15 −1.6188 Ap = « « 0 0 « ¬«0.42094 −0.027094

º ª 0 º » « » 0 −0.33868» , B p = «1.3547 » » « 0 » 0 1 » « » 0 −0.1376 ¼» ¬«0.5504 ¼»

0

0

III. HYBRID CONTROL SYSTEM DESIGN The objective is to design a hybrid control system that would convert the unstable equilibrium (2kπ,0) to a stable one and make the originally stable equilibrium ((2kπ+1),0) as a saddle point, as opposed to the phase portrait shown in Fig.2.2. The hybrid control system consists of a linear stabilizing controller, a swing-up control unit, and a supervisory coordinator. If the system is inside the region of attraction, the linear stabilizing controller will stabilize the system at x=(2kπ,0,0,0). The swing-up control unit is to be designed to move the cart strategically to give the pendulum appropriate energy to swing up to the region of attraction where the linear stabilizing controller can stabilize the pendulum and move the cart to the middle of the rail. The design of the linear stabilizing controller, the swing-up control strategy, and the supervisory control policy are briefly described as follows. A. Design of a linear digital stabilizing controller Although the inverted pendulum system is a nonlinear system, a linear stabilizing controller can still be designed to cover a large region of attraction. Since the four state

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variables of the inverted pendulum system can either be measured directly or computed easily in great precision, a linear state feedback control is our choice as the stabilizing controller. m( k ) z.o.h. u(t ) x(t ) x(k ) G p ( s) F or T PWM

where z1 (k ) represents the error to be minimized and z2 (k ) the control input constraint. An optimal state feedback controller m(k ) = Fx(k ) (3-5) is to be designed so that the closed-loop system is stable and the following cost function is minimized

(a) typical sampled-data control system

F

m(k )

Gd ( z )

J =E

x(k )

T

Q = C1u C1u ,

Since the inverted pendulum system to be controlled is an analog system and the controller will be implemented in a DSP microprocessor, which is a digital system, the closed-loop system is a sampled-data system [19]. A common practice in the design of digital controller for a sampled-data system is to find the digital equivalent of the analog system, and then design a digital controller so that the digital equivalent closed-loop system has desired performance [20, 21]. In the control system block diagram shown in Fig. 3.1(a), G p ( s) is the analog plant to be controlled, which in our case is the inverted pendulum system. The zero order hold z.o.h. [20, 21] or pulse width modulator (PWM) converts the discrete-time signal m(k) into a continuous-time u(t). On the other hand, the sampler with sampling period T converts the continuous-time signal x(t ) into a discrete-time sequence x(k ) , which is a short-hand notation for x(kT ) . The digital equivalent of G p ( s ) in Eq. (2-4) can be found as follows. Let Φ (t ) be the state transition matrix of the analog plant, i.e., ∞

T

(k ) z (k )

º »¼

(3-6)

where E[ X ] stands for the expectation of the stochastic signal X . Define

(b) discrete-time equivalent of (a) with z.o.h. Fig. 3.1 Sampled-data feedback control system.

Φ (t ) = ¦ ( Ap t ) k k !

∞

ª z «¬ k¦ =0

(3-1)

k =0

Then the combination of the zero-order-hold, the analog plant, and the sampler with sampling period T, can be replaced by the following discrete-time equivalent state-space model, Gd ( z ) : x (k + 1) = Ax(k ) + + B2 m(k ) (3-2) where T A = Φ (T ) , B2 = ª ³ Φ (T − λ )d λ º B p (3-3) ¬« 0 ¼» With the matrices Ap and B p given in Eq. (2-4), and the sampling period T = 10ms, the A and B2 matrices of the discrete-time equivalent state-space model in Eq. (3-2) can be easily found. The linear quadratic optimal control theory [3, 20] is used to design the digital state feedback controller. The state feedback controller is designed to minimize the energy of the controlled output ª z (k ) º ª C x(k ) º (3-4) z (k ) = « 1 » = « 1u » ¬ z2 (k ) ¼ ¬ D12 d m(k ) ¼

T

R = D12 d D12 d

(3-7a)

Then the state feedback gain matrix F can be computed as follows, F = −( B2T XB2 + R) −1 B2T XA (3-7b) where X is the stabilizing solution of the following, AT XA − AT XB2 ( R + B2T XB2 ) −1 B2T XA + Q = X (3-7c) With the weighting matrices chosen as C1u = 50 10 [1 1 2 2] , D12 d = 5 (3-8a)

the state feedback gain matrix F is found as F = [ -383.3 -71.51 42 71.47 ]

(3-8b)

B. Strategies for swing-up control The linear controller will stabilize the system as long as the system is in the region of attraction. However, when the system is outside the region of attraction, the linear stabilizing controller will become inadequate. The common practice to solve this problem is to find a way to swing up the pendulum back to the region of attraction. The only way to control the pendulum motion is through the cart movement. As the cart moves, one end of the pendulum attaching to the pivot will also move. The position of the pivot end of the pendulum can be precisely controlled, but controlling the angular displacement and angular velocity of the pendulum is by no means trivial. Nevertheless, some basic physics laws may shed light on developing effective strategies for swing-up. The pendulum motion is dictated by both the gravity and cart movement. It can be visualized that when the pivot end of the pendulum moves, the rest of the pendulum will lag behind because of Newton’s law of inertia. This physics law causes the other end of the pendulum always heading to the opposite direction as the pivot end moves. The second physics law that may be relevant to the swing-up is the physics of resonance. The pendulum would resonate and increase its swing amplitude if the frequency of the external influence is close to the natural frequency of the pendulum. The third relevant physics law is the conservation of energy, which has been mentioned in many swing-up inverted pendulum articles [1, 2]. The required energy for swing-up from time to time can be computed and employed to determine the control of the cart movement so that the swing-up motion can be optimized.

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To better understand how to synchronize the cart movement with the pendulum motion, we conduct simple virtual experiments in the following. First of all, the pendulum angle space is roughly partitioned into four zones as shown in Fig. 3.2. Zone I represents the region of attraction of the linear stabilizing controller. Note that the region of attraction depends not only on the angular displacement but also on the angular velocity. The partition boundaries between zones II, IV and I may change when angular velocity varies. The goal of the swing-up control algorithm is to manipulate the motion of the cart so that the pendulum in zones II, IV, and III can swing up to zone I. I

II

θ

IV III

Fig. 3.2 Partition of the pendulum angle space.

If the pendulum is in zone II and x2 > 0 (the pendulum is falling), moving the cart to the right will increase the kinetic energy of the pendulum and cause it to go down faster and then swing up higher to the other side. Contrarily, if the cart is moving to the left the two forces on the pendulum by the gravity and the cart movement are in opposite directions. This force cancellation will slow down the pendulum to not swing as high. These phenomena can be verified by the following simple computer simulation results based on the nonlinear dynamics model of the inverted pendulum derived in section II. Assume the initial condition is at ( 95D , 0, 0.5, 0), i.e., the pendulum is at the upper left of zone III and the cart is at 0.5m on the right of the rail. The pendulum will go down because of the gravity, so the cart should move to the left in order to gain more energy for the pendulum to swing up towards zone IV. This cart motion should continue until the pendulum is close to the border of zone III and zone IV. At this moment the cart should prepare to reverse the direction since the pendulum will either continue to swing up into zone IV or reverse its direction in zone III and swing back to the left. In either case, the cart should move to the right to either push the pendulum in zone IV higher or help the downward pendulum gain more momentum to swing to the left. This process will continue until the pendulum swings up back into zone I, the region of attraction. It is clear now that the first guideline for swing-up control is to utilize the above three relevant physics laws and the information of pendulum displacement and velocity to keep the cart movement synchronized with the pendulum motion. The synchronization or the right timing for the cart to switch directions ensures adding energy to the pendulum rather than subtracting it. It also can be seen that the amount of energy gained for the pendulum in each swing is determined by the travel distance and speed of the cart. Hence the second guideline

for swing-up control is to determine the travel distance and speed according to the needed energy for the pendulum. In addition to the above two considerations, it is important to ensure the cart position is always within the rail length limit and always has enough room for the next move. For example, if the next cart movement is to the left the current cart position should be on the right hand side of the rail. Therefore, the third guideline for swing-up control is to achieve precise cart position management. We will employ the regulator controller design [22] to move the cart to a preset desired position, which is determined based on the second guideline and the rail length limitation. C. Supervisory coordinator The responsibility of the supervisory coordinator is to monitor the condition of the system and to choose an operating mode according to the condition. There are three operating modes for the system, the first is the stabilizing control mode in which the system is inside the range of attraction and the linear controller is working to stabilize the system at the equilibrium point x=(2kπ, 0, 0, 0). The second operating mode is the swing-up control mode in which the system is outside the region of attraction but manageable by the swing-up control unit. The swing-up control unit is working to swing up the pendulum to enter the region of attraction and back to the stabilizing control mode. The third mode is the waiting mode in which neither the linear stabilizing controller nor the swing-up control unit is engaged. Usually the system operates either at stabilizing control mode or at the swing-up control mode. However, the supervisory coordinator may choose the waiting mode when doing nothing and waiting is better than the other two options. The supervisory coordinator needs the information of the range of attraction to define the switching surface. The range of attraction is mainly determined by the angular displacement and angular velocity of the pendulum. These two conditions are coupled. Using fixed absolute value limits on the angle and speed to define the range of attraction would unnecessarily shrink the region of attraction significantly. Therefore, the range of attraction should be represented by a two dimensional look-up table or graph. IV. COMPUTER SIMULATIONS We have conducted more than ten simulations and the corresponding experiments with a variety of initial conditions and disturbances. Due to page limitation, only two of them will be presented in the following. The closed-loop system used in simulation includes the nonlinear state-space model in section I and the hybrid control system designed based on the guidelines in section III. The two simulations only differ in initial conditions and the external disturbances. In the first simulation, the initial condition is assumed x0 = (π, 0,0,0) i.e., the pendulum is at the downward

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position, no angular velocity and the cart rests at the middle of the rail. Fig.4.1 (b) shows the displacement of the cart in blue, and the velocity in red. We can see that the cart moves back and forth to increase the pendulum swing as shown in Fig 4.1(a) until about 7.2 sec the linear stabilizing controller takes control to stabilize the system. Fig. 4.1(c) shows the torque of the motor and the time of controller switching. The torque input is limited to ±0.5N-m in the simulations. Part (d) shows the phase portrait of the pendulum. Note that at steady state the pendulum is at the upward position, -6.28 rad, which is equivalent to 0 rad. 10

(a)

x2 (rad/s)

0

-10

the DSP microprocessor controlled inverted pendulum system. The mechanical cart-pendulum hardware used is the Quanser High Fidelity Linear Cart as shown in the left picture of Fig.5.1. The hybrid control system designed based on the design concept in section III is implemented in the Texas Instruments DSP microprocessor TMS320F2812 as shown in the right picture of Fig.5.1 The sampling period is chosen as 10 ms. The hybrid control system remains the same throughout the following two experiments. The two experiments only differ in initial conditions and disturbances.

x1 (rad) 0

2

4

6

t ( sec)10

8

5

(b)

x3 (m)

0 -5

x4 (m/s) 0

2

4

6

(c)

t ( sec) 10

8

1

Fig. 5.1 The DSP microprocessor controlled inverted pendulum.

switch

10

0

0

10

4

6

t ( sec) 10

8

-10 0 1

x2 (rad/s)

(d) 0 -10

2

-6

-5

-4

-3

-2

-1

(a)

-20

-100 0 10

x2 (rad/s) 0

2

4

6

8

t ( sec)

10

2

(b)

(d) 0 x3 (m)

0 -2

2

4

6

8

t ( sec)

10

1

(c)

0

u (N-m) -1

0

5

2

4

6

8

t ( sec) 10

x2 (rad/s)

(d) 0 -5 -0.2

x1 (rad) -0.1

0

0.1

0.2

0.3

6

x4 (m/s)

2

4

8

t ( sec) 10

x3 (m)

6

8 t ( sec) 10

u (pwm duty cycle %)

0.4

2

4

6

8 t ( sec) 10

x2 (rad/s)

x1 (rad) -10 -7 -6 -5 -4 -3 -2 -1 0 Fig.5.2 Realtime results with x0 =(-π , 0, 0, 0). (a-c) time responses. (d) phase portrait, x2 vs. x1.

x4 (m/s) 0

4

(c) 0

x1 (degree)

0

2

-1 0 100

Fig.4.1 Simulation with x0 =(π , 0, 0, 0). (a-c) time responses. (d) phase portrait, x2 vs. x1. 20

x1 (rad)

(b) 0

x1 (rad) -7

x2 (rad/s)

(a) 0

u (N-m) -1

0.5

Fig.4.2 Simulation accommodation of small disturbances on the stick. (a-c) time responses. (d) phase portrait, x2 vs. x1.

Fig.4.2 shows the robustness of the system against a moderate disturbance. The system is assumed stabilized at the equilibrium. A disturbance occurs around t = 6 sec that causes the pendulum to deviate about 18 degree. The linear stabilizing controller was able to bring the system back to the equilibrium point almost right away. V. REAL-TIME EXPERIMENT RESULTS In this section, we will show the real-time performance of

In the first experiment, the initial condition is assumed x0 = (π, 0,0,0), i.e., the pendulum is at the downward position, no angular velocity and the cart rests at the center of the rail. Fig.5.2 (b) shows the displacement of the cart in blue, and the velocity in red. We can see that the cart moves back and forth around the center of the rail to increase the pendulum swing as shown in Fig. 5.2(a) until about t = 6.6sec when the linear stabilizing controller takes control to stabilize the system. Fig. 5.2(c) shows the duty cycle of the PWM control signal. Part (d) shows the phase portrait of the pendulum. Note that in (a) and (d) the steady state value −6.28 rad is equivalent to 0 rad, which represent the upward pendulum position. It is also noted that in (c) the control signal still fluctuates after the system is stabilized, which is different from those in the simulation shown in section IV where control input became 0 after achieving stabilization. In simulations, the motor dynamics is assumed linear and able to operate around 0 volt, or 0 duty cycle. However, in reality

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the motor would not move unless the magnitude of the input PWM duty cycle is greater than 25%. 20

x1 (degree)

(a) 0 -20 2

x2 (rad/s) 0

5

100

15

10

t ( sec)

15

10

t ( sec)

15

x3 (m)

(b) 0 -2

t (sec)

10

x4 (m/s) 0

5

(c) 0 -100 5

u (pwm duty cycle %) 0

5

x2 (rad/s)

(d) 0 -5 -20

x1 (rad) -15

-10

-5

0

5

10

15

20

Fig.5.3 Realtime accommodation of small disturbances on the stick. (a-c) time responses. (d) phase portrait, x2 vs. x1.

Fig. 5.3 shows the robustness of the system against moderate disturbances. The system is assumed stabilized at the equilibrium. As it can be seen in Fig.5.3a, the pendulum was hit four times, two from the left and the other two from the right, but it did not deviate much and it came right back to the equilibrium. The linear stabilizing controller is able to bring the system back to the equilibrium point almost right away. The above real-time performance graphs are compatible to the simulations graphs in section IV. Actually the real time experiment performance is slightly better than the simulation performance in terms of the swing-up time and the cart position management. Although only two scenarios were shown in the paper, the system virtually has achieved global stabilization in the sense that the pendulum will be stabilized at the (0, 0, 0, 0) despite the initial condition. The videos of the above experiments are available at the websites [23]. VI. CONCLUDING REMARKS A set of simple guidelines for the design of hybrid control system has been successfully implemented to stabilize the pendulum at the upward position for all possible initial conditions. The simulation and real-time experiment have shown that the hybrid control system can always swing up the pendulum back to the region of attraction where the system can be stabilized by the linear stabilizing controller. The system is robust under the influence of moderate disturbances. VII. REFERENCES [1] [2]

K. J. Astrom and K. Furuta, "Swinging up a Pendulum by Energy Control," Automatica, vol. 36, pp. 287-295, 2000. J. Zhao and M. W. Spong, "Hybrid control for global stabilization of the cart-pendulum system," Automatica, vol. 37, pp. 1941-1951, 2001.

[3]

H. Kwakernaak and R. Sivan, Linear Optimal Control Systems: John Wiley & Sons, Inc., 1972. [4] S. Hu and B. C. Chang, "Design of a nonlinear H_inf controller for the inverted pendulum system," in IEEE International conference on control applications, Trieste, Italy, 1998, pp. 699-703. [5] B. Srinivasan, P. Huguenin, and D. Bonvin, "Global stabilization of an inverted pendulum _ Control strategy and experimental verification," Automatica, vol. 45, pp. 265-269, 2009. [6] A. Ohsumi and T. Izumikawa, "Nonlinear Control of Swingup and Stabilization of an Inverted Pendulum," in Proceedings of the 34th IEEE Conference on Decision and Control, New Orleans, LA, 1995, pp. 3873-3880. [7] K. J. Astrom, J. Aracilb, and F. Gordillo, "A family of smooth controllers for swinging up a pendulum," Automatica, vol. 44, pp. 1841–1848, 2008. [8] M. Bugeja, "Non-Linear Swing-Up and Stabilizing Control of an Inverted Pendulum System," in EUROCON 2003, Ljubljana, Slovenia, 2003. [9] D. Angeli, "Almost global stabilization of the inverted pendulum via continuous state feedback," Automatica, vol. 37, pp. 1103-1108, 2001. [10] C. W. Tao, J. S. Taur, C. M. Wang, and U. S. Chen, "Fuzzy hierarchical swing-up and sliding position controller for the inverted pendulum–cart system," Fuzzy Sets and Systems vol. 159, pp. 2763–2784, 2008. [11] J. Yi, N. Yubazaki, and K. Hirota, "Upswing and stabilization control of inverted pendulum system based on the SIRMs dynamically connected fuzzy inference model," Fuzzy Sets and Systems vol. 122, pp. 139–152, 2001. [12] K. Furuta, M. Yamakita, and S. Kobayashi, "Swing up control of inverted pendulum," in IECON 91, 1991, pp. 2193-2198. [13] P. C. Chen, C. W. Chen, and W. L. Chiang, "GA-based modified adaptive fuzzy sliding mode controller for nonlinear systems," Expert Systems with Applications, vol. 36, pp. 5872–5879, 2009. [14] C. W. Anderson, "Learning to control an inverted pendulum using neural networks," IEEE Control Systems Magazine, pp. 31-36, April 1989. [15] Quanser, "High Fidelity Linear Cart (HFLC) User Manual." [16] TexasInstruments, "TMS320F2812 Digital Signal Processors Data Manual," Texas Instruments 2004. [17] AndvancedMotionControls, "BE25A20AC SERIES BRUSHLESS SERVO AMPLIFIERS, Model: BE25A20AC," Advanced Motion Controls. [18] A. Isidori, Nonlinear Control Systems. NY: Springer-Verlag, 1989. [19] T. Chen and B. A. Francis, Optimal sampled-data control systems. Berlin: Spring-Verla, 1995. [20] C. L. Phillips and H. T. Nagle, Digital control system analysis and design, 3rd ed.: Prentice Hall, 1995. [21] G. F. Franklin, J. D. Powell, and M. Workman, Digital control of dynamic systems: Addison Wesley, 1997. [22] B. C. Chang, G. Bajpai, and H. G. Kwatny, "A Regulator Design to Address Actuator Failures," in The 40th IEEE Conference on Decision and Control, Orlando, Florida, 2001. [23] B. C. Chang, "Systems and Control Lab http://control.mem.drexel.edu/," MEM Dept, Drexel University, Philadelphia, 2009.

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