Rotational Motion Control Design for Cart-Pendulum System with Lebesgue Sampling

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1 American Control Conference on O'Farrell Street, San Francisco, CA, USA June 9 - July, Rotational Motion Control Design for Cart-Pendulum System with Lebesgue Sampling Hiroshi Ohsaki, Masami Iwase and Shoshiro Hatakeyama Abstract This study addresses a discretization method with Lebesgue sampling for a type of nonlinear system, and proposes a control method based on the discrete system model. A system is used as this example. Applying this control method to some real system, how to implement the controller is a crucial problem. To overcome the problem, an impulsive Luenberger observer is introduced with a numerical forward mapping from the current system state to the one-step ahead state by well-known Runge-Kutta method. As the result, a system with a quantizer, whose quantization interval is relatively large, can be controlled effectively. Numerical simulations are performed to verify the effectiveness of the proposed method. I. INTRODUCTION Discrete-time control has nice properties and is natural for real systems with respect to the implementation viewpoint. This assertion is based on difficulty to realize an arbitrarily short sampling interval. That means a control law described in continuous-time systems basically cannot be implemented to real systems as it is, with the exception of analog devices use. Hence, digital control systems with a time-invariant and constant sampling interval are usually utilized to implement a desired control law by some digital devices including computers, DSPs and FPGAs []. Recently some interesting extensions on the digital control are discussed. Lebesgue sampling is one of such topics. In usual digital control, the update of the control input is performed every sampling interval, and the sampling interval is given by chopping the time axis at some regular intervals. On the other hand, in the Lebesgue sampling case, the update of the control input is performed whenever the output of the system exceeds the given levels which are decided by chopping the output range like chopping the time axis in the usual digital control case []. In other words, the control input is updated whenever some events on the output arise. In this sense, the digital control with Lebesgue sampling can be regarded as an event-based control [], []. As other related studies, a comparison between periodic and Lebesgue sampling for one-dimensional systems can be found in []. In such literature, it is shown that some impulsive control based on the Lebesgue sampling may reduce the average H. Ohsaki is with the Graduate School of Advanced Science and Technology, Tokyo Denki University, - Kanda-Nishiki-cho, Chiyoda-ku, Tokyo, -87, Japan. ohsaki@ctrl.fr.dendai.ac.jp M. Iwase and S. Hatakeyama are with the Department of Robotics and Mechatronics, School of Science and Technology for Future Life, Tokyo Denki University, - Kanda-Nishiki-cho, Chiyoda-ku, Tokyo, -87, Japan. {iwase,sho}@fr.dendai.ac.jp sampling frequency to achieve the almost same performance as the periodic sampling case. The concept of the Lebesgue sampling-based control is very natural for systems with many digital sensors such as encoders, and also for networked-systems. Typically we can say cheaper sensors are desired from the cost viewpoint for marketed products such as cars. Such cheaper sensors, however, doesn t provide good resolution in general, and a typical controller cannot achieve good performance. Hence, there are several studies on observer-based control which considers the quantization effect of low-resolution sensors to recover good control performance [], [7]. The systems via some networks are another example of Lebesgue sampling systems. The information via networks is not continuous but intermittent. The arrival intervals between previous and current information are also not fixed and varying. Hence, if a natural conception on the system via networks leads to the control law updated when the new information comes, i.e. event-based control. Montestruque and Antsaklis [8] addressed this issue and proposed an interesting model-based control for networked systems. Their and our previous methods in [7] have many points of similarity. This paper is an extension of our previous method in [7]. Especially a - system, which is nonlinear, is used as a specific example. The discretization method with Lebesgue sampling for this type of nonlinear systems are discussed first of all. The control purpose of the system is to keep the rotational speed of the, and a control. Once the discrete system is obtained, a control law based on the model is derived to realize the purpose. The control law practically can be designed by the well-known linear servo control theory [9] because the system can be represented by a linear system by the proposed discretization with Lebesgue sampling. However, applying this control method to some real system, the implementation of the controller becomes a crucial problem. To overcome the problem, according to the analogy of [7], an impulsive Luenberger observer is introduced. The impulsive Luenberger observer requires the forward mapping from the current system state to the one-step ahead state. Hence we also describe a numerical forward mapping by well-known Runge-Kutta method. As the result, a - system with a quantizer, whose quantization interval is relatively large, can be controlled effectively. Numerical simulations are performed to verify the effectiveness of the proposed method //$. AACC 87

2 y - y y_qntz The following equation of motion of only the can be extracted from ). m p r p cos θ p ẍ c + J p + m p r p) θp = m p gr p sin θ p. ) The acceleration of the, ẍ c, is regarded as the input to the system ). Jp + m p rp ) θp = m p gr p sin θ p m p r p cos θ p ẍ c. ) - Fig.. Comparison of the linear relation, y = x, with the quantizer output, qy), in the case of q n =. y O x θ p m p, J r p p x p, y p ) m c F cx x c, yc) x The following rearrangement of the angular acceleration, θ p, can hold in general. θ p = d θ p dt = dθ p d θ p = dt dθ θ d θ p p ) p dθ p Substituting ) into ) yields Jp + m p r p) θp d θ p dθ p = m p gr p sin θ p m p r p cos θ p ẍ c. ) Suppose the acceleration of the during the interval from t k and t k+, ẍ c [k], be constant. Integrating both term of ), θp [k+] Jp + m p rp ) θp d θ p Fig.. The schematic figure of a - model. θ p, x p, y p) and x c, y c ) show the angle of the, the CoG of the, and the CoG of the, respectively ṪABLE I PHYSICAL PARAMETERS AND VARIABLES OF THE CART-PENDULUM. Distance between hinge and CoG r p. [m] of Principal moment of inertia J p. [kg m ] of Mass of m p.7 [kg] Mass of m c. [kg] Angle of θ p [rad] Position of CoG of x p, y p) [m] Position of CoG of x c, y c ) [m] Input force to F xc [N] II. DISCRETIZATION OF CART-PENDULUM SYSTEM BY LEBESGUE SAMPLING A quantizer, q ), used in this study is defined as qθ) := q n round θ/q n ), ) = θ p[k] θp [k+] θ p [k] m p gr p sin θ p m p r p cos θ p ẍ c [k]) dθ p. 7) Rearranging 7), we have Jp + m p r ) p θ p [k + ] θ p[k]) = m p gr p cos θ p [k + ] cos θ p [k]) m p r p sin θ p [k + ] sin θ p [k]) ẍ c [k], 8) θ p[k + ] = θ p[k] where + αθ p [k], θ p [k + ]) + βθ p [k], θ p [k + ])ẍ c [k], 9) αθ p [k], θ p [k + ]) = m pgr p cos θ p [k + ] cos θ p [k]), βθ p [k], θ p [k + ]) = m pr p sin θ p [k + ] sin θ p [k]). where θ is an input value to be quantized, and q n is the quantization interval, and the function round ) is the Introducing a new input, u p [k], the present input to 9), the rounding function to the nearest integer. For example, Fig. acceleration ẍ c [k], can be rewritten by illustrates the quantization of y = x by the quantizer, qx), with q n =. Let us define a time when the quantizer output ẍ c [k] = u p[k] αθ p [k], θ p [k + ]). ) changes as t k. In this paper we call the time, t k, the interrupt βθ p [k], θ p [k + ]) time. Hence t k+ shows the next interrupt time. Note that Hence 9) comes to t k+ t k for all k is NOT constant. Introduce the notation to distinguish a quantized value, θ[k], from an original value, θ p[k + ] = θ p[k] + u p [k]. ) θt k ), at t k by θ[k] = qθt k )). In ), define the state as x[k] = θ p[k] and the system A - system in Fig. is used as a plant to be matrices as Φ = and Γ =. A discrete system of the system derived by Lebesgue sampling is finally controlled in this paper. Its physical parameters and variables are shown in Table I. The equations of motion of the system are given by given by [ ] ] [ ] x[k + ] = Φx[k] + Γu mc + m p m p r p cos θ p [ẍc Fcx + m m p r p cos θ p = p r p sin θ p [k]. ) p θ p. θ p gm p r p sin θ p We re interested in a class of nonlinear systems which ) can be shown by the discrete system representation ) 88

3 or a time-varying discrete system representation with the same structure with ). Unfortunately, at the moment, we cannot describe such class clearly yet. But a piston-crank model, which can present combustion engine dynamics, can be classified into this class. We also try to extend the discretization by Lebesgue sampling with multivariable case although this study just think the case only a single variable, θ p, is quantized. Those issues are our ongoing works. III. CONTROL SYSTEM DESIGN This paper considers a control task to realize constantspeed rotational motion for the of a - system. This task can be formulated by a servo control design to keep the state x[k] = θ p[k] of ) be a constant desired value. To derive the following control system, we assume that the angular velocity of the, θ p [k], can be known at each Lebesgue sampling. This implies y[k] = Cx[k] = x[k], ) with C =. Of course, this assumption is not valid for the real system. Hence, this issue will be discussed later, and can be solved by combination of some numerical integration method and impulsive Luenberger observer, which is an extension of our previous method proposed by an author [7]. Basically the control input, u p [k], in ) is designed by the well-known optimal type servo design [9]. Considering a quadratic cost function under the discrete system ); J = x[k] T Qx[k] + u[k] T Ru[k] ), ) i= the optimal state feedback control law, u[k], is given by u[k] = fx[k] with f = R + Γ T P Γ ) Γ T P Φ, ) where P is the positive symmetric matrix as the solution of the following discrete-time Riccati equation; P = Q + Φ T P Φ Φ T P Γ R + Γ T P Γ ) Γ T P Φ. ) Here, define a reference value as y r, and consider an augmented system as follows: [ ] [ ] [ ] [ ] [ x[k + ] A x[k] B = + u z[k + ] C z[k] p [k] +. 7) yr] A state feedback control law for 7), u p [k] = [ h k ] [ ] x[k], 8) z[k] leads to the optimal type servo controller for the closedloop system. The feedback gain is given by [ ] [ ] [ ] Φ I Γ h k = fφ fγ + I, 9) C where f is the optimal feedback gain in ). IV. IMPLEMENTATION During an interval from t k and t k+, u p [k] is constant and given by 8). The corresponding acceleration, ẍ c [k], is calculated by ). Note that θ p [k + ] is given as a prior information, and is available for calculation of ) because θ p [k + ] is an output of the quantizer, ), and the quantization interval of ) is known preliminarily. From ), the acceleration, ẍ c, can be represented by Jp + m prp) ) F cx + m pr p sin θ p θ p m prpg cos θ p sin θ p ẍ c = m c + m p ) ) mp r p cos θ p ). ) Therefore, once ẍ c [k] is obtained from u p [k], the horizontal force applied to the real, F cx, is derived by F cx = m c + m p m p r p cos θ p ) ) ) u p [k] αθ p [k], θ p [k + ]) βθ p[k], θ p[k + ]) m pr p sin θ p θ p + m p r pg cos θp sin θp. ) Note that F cx in ) is continuous with respect to time, and varying even though u p [k], θ p [k] and θ p [k + ] are constant during the Lebesgue sampling interval, because ) requires continuous values of θ p and θ p. As aforementioned, new measurement data, θp [k], is obtained only at the interrupt time i k, i.e. only when the quantizer output changes. In this sense, θ p [k] and θ p [k + ] are known a priori because θ p [k] is the quantized value of the original θ p. During the interrupt times, the original signals, θ p and θ p cannot be measured. So ) cannot be applied and implemented to the system directly. Hence in the following section, we propose a numerical method to solve the problem. Here we also give a remark to control the rotational direction of the. The rotational direction depends on whether define θ p [k + ] = θ p [k] + q n or θ p [k + ] = θ p [k] q n with the quantizer interval q n. A. Numerical integration of nonlinear system by Runge- Kutta method in SDC form To overcome the addressed issue on the implementation, the key is to introduce an impulsive Luenberger observer. This scheme is a kind of analogy in [7] and [8]. Such impulsive Luenberger observer, however, requires the mapping of the current system state to the one-step ahead state. That means a discrete system of the target nonlinear system is required. However, it is a difficult problem to obtain such discrete time system. We regard the difficulties is caused by the fact the input of the system affects to the system matrices in the case of the discretization for nonlinear systems. That is, the input must be known a priori over the interval for discretization. This requirements causes a circular reference problem in control system design because the system matrices are required a prior to design the input. In our case, on the other hand, the discrete system of the target nonlinear system is used only when the state estimation is updated posteriori, i.e, after each interrupt. The diagram of the proposed controller is shown in Fig.. In the following 89

4 Rearranging ) into ), k is obtained as follows k = Ā x[k] + ) A x[k] + B u) + B u = Ā I + ) ) A x[k] + ĀB + B u. 7) For simplicity let Ā I + A ) = A, ĀB + B = B k = A x[k] + B u. 8) Fig.. The diagram of the impulsive Luenberger observer-based controller for Lebesgue sampled systems. part, the detail is derived by using the - system as the example. B. Non-linear system discretization by the Runge-Kutta method Assume that input force u from t k to t k+ is constant. [ ] T Let a state vector of the system be x = x c, θ p, ẋ c, θp. Motion equations of the system ) can be changed the formula ẋ = f x, t), ) = Ax)x + Bx)u. ) ) is approximated by using the Runge-Kutta method as follows where x[k + ] = x[k] + k + k + k + k ), ) k = f x[k], t k ), k = f x[k] + k, t k + ), k = f x[k] + k, t k + ), k = f x[k] + k, t k + ). Rearranging ) into the approximated formula, k is obtained as follows k = Ax[k])x[k] + Bx[k])u. For simplicity let Ax[k]) = A, Bx[k]) = B k = A x[k] + B u. ) In a similar way k is obtained as follows k = A x[k] + ) k x[k] + ) k + B x[k] + ) k u. For simplicity let A x[k] + k ) = Ā, Bx[k]+ k ) = B k = Ā x[k] + ) k + B u. ) In a similar way k, k are obtained as follows k = A x[k] + B u, 9) A = Ā I + ) A, Ā = A x[k] + ) k, k = A x[k] + B u, ) A = Ā I + A ), Ā = A x[k] + k ). Thus a discretized formula of ) can be described as follows x[k + ] = Φ x[k], ) x[k] + Γ x[k], ) u, ) Φ x[k], ) = I + A + A + A + A ), Γ x[k], ) = B + B + B + B ). C. Impulsive Luenberger Observer In the system if the angular velocity θ p can not measure then θ p need to be estimated. Consequently θ p is estimated by using ILO which consider the quantization of the angle. ILO is defined as follows { ˆxt) = f ˆxt), t, u) t / {t k } k= ˆxt) ˆxt) + L k C xt) ˆxt)) t {t k } ) k= [ ] T where ˆx is estimated state, and x = x c, q θ p ), ẋ c, θp. Assume that the angle qθ p ) and the position x c can be masured, coefficient matrix of the output equation is as follows C = [ ]. The obserer gain L k can be obtained so that all eigenvalues of Φx[k], ) L k CΦx[k], ) are inside the unit disc. V. NUMERICAL SIMULATION In the following simulation the control input and the estimated states can only be updated at the quantizer transition time. The quantization width q n = π/8[deg]. Setting the constant reference y r = π) in the system 7), the angular velocity of the system is controlled to the constant velocity θ p = π[rad/sec]. At first simulation results with the measurement velocity θ p by using the servo control law are shown. 8

5 Next simulation results with only the position x c and the angle θ p are shown. In this results the other states is estimated by using ILO ) with the discrete system ). A. Angular velocity control with the measurement θ p In this simulation integral calculation function is rkf) with MaTX Windows9x/ME/NT//XPVisual C++ ) version..7. Step size for rkf) is. Initial condition of the angle is π/8[rad], the velocity is θ p =.7[rad/sec], otherwise. Weight matrix are set to Q = [] and R = [] for x[k] and u[k]. 7 Fig The angular velocity of the. error Fig.. The angular velocity error from constant reference π[rad/sec]. input [N] theta_p [rad] Fig.. The input to the Fig. 7. The angle of the car. From Fig. and Fig., the velocity θ p achieve the constant reference π[rad/sec]. From Fig. 7, the angle monotonic increase by the velocity which achieve the constant reference. From Fig., the input force to the updates at the quantizer transition time. From Fig. 8 and Fig. 9, the position and speed change with xc [m] Fig. 8. The position of the car xc_dot [m/sec] Fig. 9. The velocity of the car. the input force which designed the above. B. Angular velocity control with ILO In this simulation integral calculation function is rkf) with MaTX Windows9x/ME/NT//XPVisual C++ ) version..7. Step size for rkf) is. Initial condition of the angle is π/8[rad], the velocity is θp =.7[rad/sec], otherwise. Weight matrix are set to Q = [] and R = [] for x[k] and u[k]. The observer gain L k is designed discrete-time linear quadratic regulator ) with controllable pair Φ x[k], ) T, CΦ x[k], )) T. Weight matrix are set to Q o = diag,,, ) and R o = diag, ) for xt k ) and the correction term. 7 _xh Fig.. The angular velocity of the with ILO. Red solid line is the angular velocity θ p. Orange solid line is the estimated angular velocity. error Fig.. The angular velocity error from constant reference π[rad/sec] with ILO. 8

6 input [N] theta_p [rad] Fig.. xc [m] 8 Fig.. The input to the with ILO. _xh Fig.. xc_dot [m/sec] The angle of the car with ILO. _xh The position of the car with ILO. _xh Fig.. The velocity of the car with ILO. From Fig. and Fig., the velocity θ p achieve the constant reference π[rad/sec]. From Fig. to Fig., the input force, estimated states are updated at the quantizer transition time. The estimated states achieve the real states. VI. DISCUSSION In the control system design, we assume that measurement states are changed when the quantizer output changed. The other states are estimated by using the measured states at the same time. Inputs to the system are determined by the measured states and the estimated states. The above - model don t consider viscous friction at the joint of the. In order to consider effect of the viscous friction to discretization by Lebesgue sampling, we derive - model that includes the viscous friction force Cp θ p Nm at the joint of the. The equation of motion of the pedulum with the viscous friction force is given by m p r p cos θ p ẍ c + J p + m p r p) θp = m p gr p sin θ p C p θp. ) Now we focus on right term of ), C p θp. In order to discretize by Lebesgue sampling, both term of ) are integrated same way as 7). However the integrated right term of ), θ p [k+] θ p[k] C p θp dθ p, can NOT be solved analytically because θ p is NOT constant. Therefore the discretization of system by Lebesgue sampling can be performed for the system can be solved analytically same way as 7). VII. CONCLUSION In this paper, first of all, a discretization with Lebesgue sampling has been considered for a type of nonlinear system such as a - system. For example the system was converted into the corresponding discrete system ), which was a time-invariant linear or time-varying linear system, by the proposed method. Hence many current control design schemes can be applied to the discrete system. On the other hand, the implementation of the controller was a crucial problem, and to overcome this problem, in this paper, an impulsive Luenberger observer was introduced. This observer require basically the mapping of the current state to the one-step ahead state of the nonlinear system, and then a numerical integration method based on Runge-Kutta was also derived to give such mapping. Hence the controller implemented was given by the combination of the impulsive Luenberger observer and the numerical integration method. With the controller, the output of the closedloop system was controlled to be a desired value even though the controller worked only when the quantization occurred. The numerical simulation showed the effectiveness of the proposed system. As future works, we re now interested in the class of nonlinear systems to which the proposed system can be applied. A combustion engine piston-crank model might be an example. REFERENCES [] K. J. Åström, K. Johan and B. Wittenmark, Computer-controlled systems. third edition, Prentice Hall, 997. [] K. J. Åström and B. M. Bernhardsson, Comparison of Riemann and Lebesgue sampling for first order stochastic systems, In Proceeding of the th IEEE Conference on Decision and Control, Las Vegas, Nevada USA, December, pp.. [] K. E. Årzén, A simple event-based PID controller, In Proceeding of the th IFAC World Congress, Beijin, China, 999. [] N. Marchand, Stabilization of Lebesgue sampled systems with bounded controls: the chain of integrators case, In Proceedings of the 7th IFAC World Congress, Seoul, Korea, 8, pp. 7. [] Y. K. Xu and X. R. Cao, Time Aggregation Based Optimal Control and Lebesgue Sampling, In Proceedings of the th IEEE Conference on Decision and Control, New Orleans, LA, USA, December 7, pp [] J. Sur and B. Paden, Observers for Linear Systems with Quantized Outputs, In Proceedings of the American Control Conference, Albuquerque, New Mexico, 997, pp.. [7] M. Iwase, S. Katohno and T. Sadahiro, Stabilizing Control for Linear Systems with Low Resolution Sensors, 8th IEEE International Conference on Control Applications Part of 9 IEEE Multi-conference on Systems and Control, Saint Petersburg, Russia, July 8, 9, pp. 9. [8] Luis A. Montestruque and Panos J. Antsaklis, On the Model-Based Control of Networked Systems, Automatica, vol., no., pp ) [9] T. Mita, Optimal Digital Feedback Control Systems Counting Computation Time of Control Laws, IEEE Transactions on Automatic Control, AC, 98, pp. 8. 8

Rotational Motion Control Design for Cart-Pendulum System with Lebesgue Sampling

Journal of Mechanical Engineering and Automation, (: 5 DOI:.59/j.jmea.. Rotational Motion Control Design for Cart-Pendulum System with Lebesgue Sampling Hiroshi Ohsaki,*, Masami Iwase, Shoshiro Hatakeyama