Optimal Control Using State-dependent Riccati Equation (SDRE) for a Hydraulic Actuator

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Proceedings of the World Congress on Engineering 2013 Vol III, WCE 2013, July 3 - 5, 2013, London, U.K. Optimal Control Using State-dependent Riccati Equation (SDRE) for a Hydraulic Actuator F. Liccardo, S. Strano and M. Terzo  Abstract— This paper presents a nonlinear control for a hydraulic actuator with significant nonlinearities. These are the key factors causing delay and error in the hydraulic actuation response and highly limit the performances of the classical linear control. Hence, a nonlinear control design based on the mathematical model of the hydraulic actuator is employed. The proposed approach consists of a nonlinear feedback control based on the state-dependent Riccati equation (SDRE). The control performance is demonstrated by both simulations and real-time experiments. The experimental results validate the proposed approach and highlight a good accordance with simulations. Index Terms— Hydraulic actuator, optimal control, SDRE, Riccati equation, nonlinear, tracking, real-time. I. INTRODUCTION HYDRAULIC actuators employ hydraulic pressure to drive an output member. These are used where high speed and large forces are required. The fluid used in hydraulic actuator is highly incompressible so that pressure applied can be transmitted instantaneously to the attached member. Hydraulic components, because of their high speed and pressure capabilities, can provide high power output with small weight and size in comparison to electric system components. Hydraulic actuators can be found in transportations, industrial machineries, seismic applications [1], and earth moving equipments. However, the dynamics of hydraulic systems are highly nonlinear [2] due to the pressure-flow rate relationship, the dead band of the control valve and the frictions. These nonlinearities highly limit the performance achieved by the classical linear controller. In the past, much of the work in the control of hydraulic systems has used linear model [3] or local linearization of the nonlinear dynamics about the nominal operating point [4]. Suitable adaptive approaches are employed when there is no knowledge of the parameter values [5], [6]. In order to take system uncertainties into account, robust approaches can be adopted [7], [8]. In [9], a sliding mode control F. Liccardo is with the Dipartimento di Ingegneria Industriale, Università degli Studi di Napoli Federico II, 80125 ITALY (e-mail: felice.liccardo@unina2.it) S. Strano is with the Dipartimento di Ingegneria Industriale, Università degli Studi di Napoli Federico II, 80125 ITALY (corresponding author, phone: +390817683277; e-mail: salvatore.strano@unina.it). M. Terzo is with the Dipartimento di Ingegneria Industriale, Università degli Studi di Napoli Federico II, 80125 ITALY, (e-mail: m.terzo@unina.it). applied to an asymmetric single-rod cylinder was presented. In this paper a SDRE-based control for a hydraulic actuator is proposed and applied to a hydraulic cylinder of a seismic test bench. The basic idea of the SDRE technique is to capture the nonlinearities by bringing the nonlinear system to a linear structure having state-dependent coefficient (SDC) matrices, and minimizing a nonlinear performance index having a quadratic-like structure [10]. The suboptimal control action can be obtained solving online an algebraic Riccati equation (ARE) using the SDC matrices [11]. This paper continues the work done in [12] and [13] and shows experimental results in order to validate the proposed nonlinear approach. The rest of the paper is organized as follows: a description of the proposed control is given in Section II. In Section III the nonlinear model of the hydraulic actuator is derived and in Section IV the control has been particularized the specific system. Simulation results are reported in Section V and in Section VI the main experimental results are presented. II. SDRE FORMULATION Consider a nonlinear observable systems represented in general form by equations x  f (x, u) (1) where x  n and u  m are the state and the control respectively. Assume that the origin is an equilibrium point and suppose that the dynamic model of the system can be placed in the SDC form x  A(x)x  B(x)u . (2) Consider the autonomous, infinite-horizon, nonlinear regulator problem of minimizing the performance index   J  1  xT Qx  uT Ru dt 20 (3) with respect to the state x and the control u, subject to the nonlinear differential constraints (2); where Q  0 and R  0 are symmetric weighting matrices. The SDRE control method provides an approximate nonlinear feedback solution of the above problem. The feedback gain equation is given as ISBN: 978-988-19252-9-9 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) WCE 2013 Proceedings of the World Congress on Engineering 2013 Vol III, WCE 2013, July 3 - 5, 2013, London, U.K. K (x)  R 1B T (x)P(x) (4) V0 2β PL  Ap y  QL (9) where P(x) is the symmetric, positive-definite solution of the SDRE of the form AT (x)P(x)  P(x) A(x)  Q  P(x)B(x)R1BT (x)P(x)  0 (5) The SDRE controller can be implemented as a servomechanism, similar to the that of a linear quadratic regulator [14]. Given a desired state trajectory xd , the SDRE servo control action is then given by uSD  R1BT (x)P(x)(x  xd ) . (6) where V0 is the volume of each chamber for the centered position of the piston, QL= (QA+QB)/2 is the load flow and β is the effective Bulk modulus. An overlapped four-way valve is considered: typically, this kind of valve is characterized by the lands of the spool greater than the annular parts of the valve body. Consequently, the flow rate is zero (dead band) when the spool is in the neighbourhood of its central position The load flow depends on the supply pressure, the load pressure and the valve spool position in accordance with the following QL  Ψ (ve )ve Ps  PL (10) III. DYNAMICAL MODEL OF THE HYDRAULIC ACTUATION SYSTEM In order to describe the behaviour of the system, a set of differential equations is derived in the following. The hydraulic actuator under consideration consists of a doubleended hydraulic cylinder driven by a four-way spool valve. The hydraulic cylinder is coupled to a mass that moves on linear guides (Fig. 1). For the derivation of the mathematical model, the following hypothesis have been adopted: a) fluid properties not depending on the temperature; b) equal piston areas; c) equal oil volume for each side (with the barrel in a central position); d) negligible internal and external fluid leakages; e) tank pressure equal to zero. The dynamics of the movable mass displacement (y) is governed by my  σy  Ff ( y)  Ap PL , (7) where m is the mass of the load, σ is the viscous friction coefficient, Ff is the friction force, Ap is the piston area, PL = PA-PB is the load pressure, PA and PB are the pressures inside the two chambers of the cylinder. The friction force is represented by the following equation Fc sgn( y)  μmg sgn( y) Ff ( y )   Z  Z  Fc0  μ0mg  y  0 y  0 (8) where Fc is the Coulomb friction force in the hydraulic actuator (assumed equal to its static value Fc0), μ is the Coulombian friction coefficient of the linear guides (assumed equal to its static value μ0), g is the gravitational acceleration and Z is the net tangential force that acts on the actuator when it is not moving [15]. The load pressure dynamics [2] is given by where ve is the displacement signal of the spool valve and Ψ(ve) is a variable gain functional to describe the valve dead band. The analytical expression of Ψ (ve) can be assumed as  kqp 1  vep ve  Ψ (ve )  0  kqn  1  ven ve  ve  vep ven  ve  vep ve  ven (11) where the pairs (ven, vep) and (kqn, kqp) are the dead band widths and the gains for the positive and negative spool displacement, respectively. The dead band nonlinearity is among the key factors causing delay and error in the hydraulic actuation response. The proportional valve dynamics can be well represented by a second order differential equation [16] ve ω2 nv  2ξv ωnv ve  ve  keu  ve0 (12) where parameters ωnv and ξv are the natural frequency and the damping ratio of the valve respectively, ve0 is the spool position bias, ke is the input gain and u is the valve command. ISBN: 978-988-19252-9-9 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) WCE 2013 Proceedings of the World Congress on Engineering 2013 Vol III, WCE 2013, July 3 - 5, 2013, London, U.K. Fig. 1. Schematic diagram of the hydraulic actuation system. Finally, the equations governing the dynamics of the whole system (movable mass + hydraulic system) are my  σy  Ff ( y)  Ap PL  V0   2β PL   Ap y  QL QL  Ψ (ve )ve Ps  PL ve  ω 2 nv ve  2ξ v ω nv ve  ω2 nv (keu  ve0 ) (13) The developed fifth order model fully describes the nonlinear dynamical behavior of the hydraulic actuation system and takes the nonlinear friction forces and the nonlinear flow rate distribution into account. The system (13) can be written in the following form y    σ m y  Ff ( y) mv~e ve  Ap PL m  PL    2βAp V0 y  2βΨ (v~e ) Ps  PL V0 ve  ve  2ξ v ω nv ve  1 ve0 v~e ωn2vve  ωn2v keu (16) IV. PARAMETERIZATION OF THE SDRE CONTROL This section explains the design of the SDRE controller. The SDC parameterization has been performed using the highly nonlinear equations (16). The matrices in (2) have been chosen as follows:    σ m 0 Ap m 0  Ff ( y) mv~e    1 0 0 0 0 A( x)    2βAp 0 0 0 2βΨ (v~e ) Ps  PL    V0 V0   0   0 0 0  2ξvωnv 00 1  1  ve0 v~e ω2nv 0     (17)   y    σ m y  Ff ( y) mve ve  Ap PL m  PL    2βAp V0 y  2βΨ (ve ) Ps  PL V0 ve ,  ve  2ξvωnvve  1 ve0 ve ω2nvve  ω2nvkeu (14) The system (14), represented in the state space form, is nonlinear in the state, autonomous and characterized by a fully known state by means of measurements. It is possible to note in the first and the third equation of (14) a division by the variable ve, as well as happens in (11). In order to prevent divisions by zero, has been introduced a functional variable v~e  ve  ε , where ε is given by: ε  0ve ve  0 otherwise (15) where the value of ve is chosen less than the minimum acceptable measurement error of the valve spool position ve. Substituting the variable v~e into the fractions of (14) and (11) follows: where the state vector is x  y y PL ve ve T . The matrix B does not depend on the state and is given by: 0   0   B 0 .   ω 2 nv k e     0   (18) For the specific problem, without loss of generality, the goal of the tracking control is that the movable mass follows a desired displacement yd(t); therefore, the weighting matrices in (3) become: Q  diag(0,qSD,0,0,0) and R  rSD , where qSD and rSD are two weighting coefficients. V. SIMULATION RESULTS In this section some simulation results, concerning the application of the proposed controller to a particular hydraulic actuation system presented in Section VI, are reported. The physical parameter values of the hydraulic actuator are: m  441 kg , σ  23555 Ns , m FC  FC0  950 N, μ  μ0  0.01, N  4326 N , Ap  0.01m2 , ISBN: 978-988-19252-9-9 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) WCE 2013 Proceedings of the World Congress on Engineering 2013 Vol III, WCE 2013, July 3 - 5, 2013, London, U.K. V0  0.004 m3 , β  1e9 Pa , kqp  6.15e  7 m3 1, s  V  Pa 2 kqn  5.86e  7 m3 1 , vep  0.43 V , ven  0.21V , s  V  Pa 2 ve0  0.01V , ke  0.49 , ωnv 152.30 , PS  6e6Pa . These parameters have been obtained with a parameter identification technique [13]. The controller is used for tracking sinusoidal motion trajectory characterized by an amplitude of 0.04 m and a frequency of 1 Hz. The simulation results reported in Fig. 2 consist of comparisons between the target and the effective movable mass displacement, the tracking error ( y  yd ) and the control action. The isolator under test (C) is located between the sliding table and a slide that can move vertically with respect to the horizontal reaction structure (D). A hydraulic jack (E) is positioned between the vertical reaction structure (F) and the slide in order to make the isolator under test vertically loaded. The supply circuit of the hydraulic actuator is mainly constituted by an axial piston pump, powered by a 75 kW AC electric motor, a pressure relief valve and a four waythree position proportional valve. The pump is characterized by a variable displacement and it is able to generate a maximum pressure of 210 bar and a maximum flow rate equal to 313 l/min. Fig. 3. Experimental test rig. Fig. 2. Simulation results of the nonlinear SDRE-based control. The maximum value of the tracking error is about 0.008 m. The difference between the peak to peak amplitude of the effective displacement and the target one is about 0.001 m, and the actuation system phase lag is of 0.03 s. To verify the prediction capacity of simulation, the results reported in Fig. 2 will be compared with the experimental ones. VI. REAL-TIME EXPERIMENTS In order to test the effectiveness of the proposed nonlinear tracking controller, experimental studies are conducted on the hydraulic actuation system of the test rig shown in Fig. 3. A. The experimental test rig The experimental test rig is a machine utilized to perform shear tests on seismic isolators [17], [18]. The actuation system consists of a double-ended hydraulic actuator placed between a fixed base and a sliding table (1.8 m x 1.59 m). The hydraulic actuator is constituted by a mobile barrel, integral with the sliding table (A), and two piston rods linked to the fixed base (B) (Fig. 3). The maximum horizontal force is 190 kN, the maximum speed is 2.2 m/s and the maximum stroke is 0.4 m (± 0.2 m). The removal of the reaction structures allows the testing machine to be used as a shaking table [19]. The full-state feedback has been obtained with the following measurements: - table position by means of magnetostrictive position sensor (FS = 0.4 m - estimated uncertainty = ± 1.2e-4 m); - pressure in the two chambers of the hydraulic cylinder by means of strain gauge sensor (FS = 400 bar - estimated uncertainty = ± 1 bar); - valve spool position by means of built-in LVDT sensor; A dSPACE DS1103 controller board, equipped with a 16-bit A/D and D/A converter, has been used for the realtime experiments. All experiments have been conducted with a sample frequency of 1 kHz. To attenuate the influence of the noise, all measured signals are processed through a low-pass filter. The supplied pressure has been fixed to 6e6 Pa. B. Experimental Results The experiments on the seismic isolator testing machine have been conducted without the isolator to be characterize. Hence, the hydraulic actuation system has been utilized only for the sliding table positioning. For sake of comparison, the experimental validation of the proposed control has been obtained with the same target displacement imposed to the movable mass in the simulations. The problem of computing the SDRE feedback gains ISBN: 978-988-19252-9-9 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) WCE 2013 Proceedings of the World Congress on Engineering 2013 Vol III, WCE 2013, July 3 - 5, 2013, London, U.K. reduces to solving (5). The proposed approach is based on finding the eigenvalues of the associated Hamiltonian matrix [20] H (x)    A(x)   Q  B(x)R1BT (x)  AT (x)   . (19) The SDRE-based feedback gains have been obtained with a pole placement algorithm in terms of the stable eigenvalues of H(x). The SDRE feedback action has been implemented as a Ccode function downloaded to the controller board and implemented in real-time. In Fig. 4 are reported the experimental results in terms of the target and the experimental effective table displacement, the tracking error and the control action. Fig. 4. Experimental results of the nonlinear SDRE-based control. Analyzing the experimental results, it is possible to assert that the maximum amplitude error is equal to 3e-5 m and the actuator phase lag is equal to 0.035 s. Concerning the tracking error, its maximum value is equal to 0.009 m. The experimental results agrees with the results predicted in simulation environment. It has been experimentally demonstrated that the proposed nonlinear controller can achieve very good performance in terms of tracking control and stability. VII. CONCLUSION In this paper has been proposed an optimal control using the SDRE for a hydraulic actuator. A fifth order dynamic model of the system has been derived taking into account the typical nonlinearities of the hydraulic actuators. The parameterization of the SDRE feedback control has been obtained directly from the fifth order model. The real-time implementation of the SDRE nonlinear optimal problem has been performed with a pole placement algorithm in terms of the stable eigenvalues of the Hamiltonian matrix. Experiments have been conducted on the hydraulic actuation system equipped on a seismic isolator test rig. The experimental results validate the proposed approach and highlight a enfocar la importancia de la producción mediática de los niños en su descubrimiento del mundo, sobre todo utilizando el periódico escolar y la imprenta. Asimismo las asociaciones de profesores trabajaron en esta línea e incluso la enseñanza católica se comprometió desde los años sesenta realizando trabajos originales en el marco de la corriente del Lenguaje Total. Páginas 43-48 45 Comunicar, 28, 2007 En el ámbito de los medios, también desde el principio del siglo XX hay ciertas corrientes de conexión. Pero es a lo largo de los años sesenta cuando se constituyeron asociaciones de periodistas apasionados por sus funciones de mediadores, que fomentaron la importancia ciudadana de los medios como algo cercano a los jóvenes, a los profesores y a las familias. Así se crearon la APIJ (Asociación de Prensa Información para la Juventud), la ARPEJ (Asociación Regional de Prensa y Enseñanza para la Juventud), el CIPE (Comité Interprofesional para la Prensa en la Escuela) o la APE (Asociación de Prensa y Enseñanza), todas ellas para la prensa escrita Estas asociaciones fueron precedidas por movimientos surgidos en mayo de 1968, como el CREPAC que, utilizando películas realizadas por periodistas conocidos, aclaraba temas que habían sido manipulados por una televisión demasiado próxima al poder político y realizaba encuentros con grupos de telespectadores. cipio del siglo XX, y nos han legado textos fundadores muy preciados, importantes trabajos de campo y muchos logros educativos y pedagógicos. La educación en medios ha tenido carácter de oficialidad de múltiples maneras, aunque nunca como una enseñanza global. Así la campaña «Operación Joven Telespectador Activo» (JTA), lanzada al final de los años setenta y financiada de manera interministerial para hacer reflexionar sobre las prácticas televisuales de los jóvenes, la creación del CLEMI (Centro de Educación y Medios de Comunicación) en el seno del Ministerio de Educación Nacional en 1983, la creación de la optativa «Cine-audiovisual» en los bachilleratos de humanidades de los institutos en 1984 (primer bachillerato en 1989) y múltiples referencias a la educación de la imagen, de la prensa, de Internet. La forma más visible y rápida de evaluar el lugar de la educación en medios es valorar el lugar que se le ha reservado en los libros de texto del sistema educa- 2. Construir la educación en los medios sin nombrarla El lugar que ocupa la edu- La denominación «educación en medios», que debería cación en los medios es muy ambiguo, aunque las cosas están cambiando recientemente. entenderse como un concepto integrador que reagrupase todos los medios presentes y futuros, es a menudo percibida En principio, en Francia, co- por los «tradicionalistas de la cultura» como una tendencia mo en muchos otros países, la educación en los medios no es hacia la masificación y la pérdida de la calidad. una disciplina escolar a tiempo completo, sino que se ha ido conformado progresivamente a través de experiencias y reflexiones teóricas que han tivo en Francia. Una inmersión sistemática nos permi- permitido implantar interesantes actividades de carác- te constatar que los textos oficiales acogen numerosos ter puntual. Se ha ganado poco a poco el reconoci- ejemplos, citas, sin delimitarla con precisión. miento de la institución educativa y la comunidad es- colar. Podemos decir que ha conquistado un «lugar», 3. ¿Por qué la escuela ha necesitado casi un siglo en el ámbito de la enseñanza transversal entre las dis- para oficilializar lo que cotidianamente se hacía en ciplinas existentes. ella? Sin embargo, la escuela no está sola en esta aspi- Primero, porque las prácticas de educación en me- ración, porque el trabajo en medios es valorado igual- dios han existido antes de ser nombradas así. Recor- mente por el Ministerio de Cultura (campañas de foto- demos que no fue hasta 1973 cuando aparece este grafía, la llamada «Operación Escuelas», presencia de término y que su definición se debe a los expertos del colegios e institutos en el cine ), así como el Minis- Consejo Internacional del Cine y de la Televisión, que terio de la Juventud y Deportes que ha emprendido en el seno de la UNESCO, definen de esta forma: numerosas iniciativas. «Por educación en medios conviene entender el estu- Así, esta presencia de la educación en los medios dio, la enseñanza, el aprendizaje de los medios moder- no ha sido oficial. ¡La educación de los medios no apa- nos de comunicación y de expresión que forman parte rece oficialmente como tal en los textos de la escuela de un dominio específico y autónomo de conocimien- francesa hasta 2006! tos en la teoría y la práctica pedagógicas, a diferencia Este hecho no nos puede dejar de sorprender ya de su utilización como auxiliar para la enseñanza y el que las experiencias se han multiplicado desde el prin- aprendizaje en otros dominios de conocimientos tales Páginas 43-48 46 Comunicar, 28, 2007 como los de matemáticas, ciencias y geografía». A pe- mente en todas las asignaturas. Incluso los nuevos cu- sar de que esta definición ha servido para otorgarle un rrículos de materias científicas en 2006 para los alum- reconocimiento real, los debates sobre lo que abarca y nos de 11 a 18 años hacen referencia a la necesidad no, no están totalmente extinguidos. de trabajar sobre la información científica y técnica y En segundo lugar, porque si bien a la escuela fran- el uso de las imágenes que nacen de ella. cesa le gusta la innovación, después duda mucho en Desde junio de 2006, aparece oficialmente el tér- reflejar y sancionar estas prácticas innovadoras en sus mino «educación en medios» al publicar el Ministerio textos oficiales. Nos encontramos con una tradición de Educación los nuevos contenidos mínimos y las sólidamente fundada sobre una transmisión de conoci- competencias que deben adquirir los jóvenes al salir mientos muy estructurados, organizados en disciplinas del sistema educativo. escolares que se dedican la mayor parte a transmitir Este documento pretende averiguar cuáles son los conocimientos teóricos. La pedagogía es a menudo se- conocimientos y las competencias indispensables que cundaria, aunque los profesores disfrutan de una ver- deben dominar para terminar con éxito su escolaridad, dadera libertad pedagógica en sus clases. El trabajo seguir su formación y construir su futuro personal y crítico sobre los medios que estaba aún en elaboración profesional. Siete competencias diferentes han sido te- necesitaba este empuje para hacerse oficial. nidas en cuenta y en cada una de ellas, el trabajo con Aunque el trabajo de educación en los medios no los medios es reconocido frecuentemente. Para citar esté reconocido como disciplina, no está ausente de un ejemplo, la competencia sobre el dominio de la len- gua francesa definen las capa- cidades para expresarse oral- La metodología elaborada en el marco de la educación en mente que pueden adquirirse con la utilización de la radio e, medios parece incluso permitir la inclinación de la sociedad incluso, se propone fomentar de la información hacia una sociedad del conocimiento, como defiende la UNESCO. En Francia, se necesitaría unir el interés por la lectura a través de la lectura de la prensa. La educación en los medios las fuerzas dispersas en función de los soportes mediáticos y orientarse más hacia la educación en medios que al dominio adquiere pleno derecho y entidad en la sección sexta titulada «competencias sociales y cívi- técnico de los aparatos. cas» que indica que «los alum- nos deberán ser capaces de juz- gar y tendrán espíritu crítico, lo que supone ser educados en los las programaciones oficiales, ya que, a lo largo de un medios y tener conciencia de su lugar y de su influencia estudio de los textos, los documentalistas del CLEMI en la sociedad». han podido señalar más de una centena de referencias a la educación de los medios en el seno de disciplinas 4. Un entorno positivo como el francés, la historia, la geografía, las lenguas, Si nos atenemos a las cifras, el panorama de la las artes plásticas : trabajos sobre las portadas de educación en medios es muy positivo. 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