Virtual Power Limiter System which Guarantees Stability of Control Systems

Transcription

1 Virtual Power Limiter Sstem which Guarantees Stabilit of Control Sstems Katsua KANAOKA Department of Robotics, Ritsumeikan Universit Shiga , Japan Abstract In this paper, Virtual Power Limiter Sstem is proposed. This makes it possible to guarantee the stabilit of almost all sstems which include unknown characteristics: flexibilit in flexible manipulators, compressibilit in pneumatic servo sstems, human dnamics in man-machine sstems, hardware nonlinearit in mechatronic sstems, and so on. The details of the proposed limiter sstem and some case studies are described. I. INTRODUCTION It is ver important problem to guarantee the stabilit of feedback sstems. Unknown characteristics in the sstems badl affect to the stabilit. Some approaches are attempted to guarantee the stabilit and can improve performances, such as adaptive or robust based on dnamics modeling and identification. In the case that the unknown characteristics cannot be estimated at all, however, the onl solution seems to change the sstems more. This is usuall based on trial and error, so it is ver difficult to strike a balance between the stabilit and high performance. Unlike the above solution, we proposed Passivit Monitor and Software Limiter [1], which is the framework to stabilize unknown sstems without an model inside. In this paper, we redefine it as Virtual Power Limiter Sstem (VPLS) in more sophisticated and more general form. The problem which we should solve is as follows. Assume that a sstem as shown in Fig.1 is used b a certain user. We call it User s Control Sstem (UCS). The user s and the user s in the UCS are defined as in Fig.1. We the designers of the VPLS do not necessaril know the details of the UCS, except the output (t) R n from the UCS at time t. Here, our problem is how to guarantee the stabilit of the UCS which can go unstable easil b unknown factors, and how to strike a balance between the stabilit and high performance. This solution is given b adding the VPLS to the UCS. Next, the is connected l to the UCS as shown in Fig.2. A is defined as the sstem which evaluates the stabilit of the UCS b ing the through this input-output connection. A limiter is defined as the sstem which regulates the and guarantees the stabilit of the UCS b tuning some parameters in the UCS based on the. The VPLS is composed of the and the limiter. The details of the VPLS are shown below. A. Virtual Power Monitor Fig.2 is the UCS to which the sstem is added. R n is the input from the. should make a conjugate pair with the output. This does not necessaril need to be input to the user s. The connection of into the UCS is usuall. The ed value in the is defined as the following. (t) = T (t) (t) (1) This means the transferred from the to the UCS. The stabilit of the UCS is evaluated b this from the. B. Virtual Power Limiter The limiter guarantees the stabilit of the UCS b tuning the UCS based on the ed. Fig.3 is the conceptual scheme of the limiter. II. VIRTUAL POWER LIMITER SYSTEM In this section, the limiter sstem is defined. First, the VPLS needs a expected to the user s stabl, separatel from the user s. The into this will be the same for the user s, or more. Fig. 1. input led User s Control Sstem (UCS)

2 III. VPLS DESIGN BASED ON CONCEPT OF PASSIVITY In this section, the VPLS is designed based on the concept of passivit. Then, it is shown theoreticall that the VPLS guarantees the stabilit of the UCS. A. Passivit The concept of passivit [2] is one of the most important characteristics of robot sstems. The definition of the passivit is given. In terms of the input u R n and the output, the sstem is called passive if the following inequalit is alwas satisfied, T (τ)u(τ) dτ γ 2 ( t > ) (2) where γ 2 is a non-negative constant onl depending on the initial state of the sstem. Phsicall, it is the initial energ of the sstem. B. Stabilit Evaluation In order to evaluate the stabilit of the UCS b the limiter, the value E v calculated from is considered. E v (t) = (τ) dτ = T (τ) (τ) dτ (3) This E v means the energ transferred from the to the UCS. The following inequalit is introduced to evaluate the stabilit of the UCS. E v (t) = T (τ) (τ) dτ E v (4) where E v is a certain arbitrar constant which the designer should define. This inequalit is considered as a stretch of the concept of passivit (2). In other words, b ing the input and output from t = to t, the UCS cannot be distinguished from a passive phsical sstem with the initial energ E v. Suppose that the following negative feedback is chosen as the. sstem = K( d ) (5) K R n n is a positive diagonal matrix, and d R n is a to the. Then, = d T dτ = If (4) is satisfied, T dτ + d T dτ E v + (K 1 + ) T dτ T K 1 dτ (6) T K 1 dτ (7) the UCS with the VPLS apparentl satisfies outputdissipativit. Equation (4) is a sufficient condition for the stabilit of the UCS with the VPLS, so the stabilit of the UCS can be evaluated b ing and (4). C. Stabilit Analsis of UCS with VPLS Here the stabilit of the UCS with the VPLS is analzed using Popov s Hper-Stabilit Theorem. The UCS is considered as the backward time-variant nonlinear block, and the is considered as the forward timeinvariant linear block. According to Popov s Hper-Stabilit Theorem, the sufficient conditions for the asmptotic stabilit of the feedback sstem of the UCS and the are: 1) the is time-invariant and strictl positive real for the input and the output, 2) the UCS satisfies (4) for the input and the output. These sufficient conditions are restated as the design polic of the VPLS: 1) the is designed to be timeinvariant and strictl positive real for the input and the output, 2) the VPLS can tune some parameters in the UCS and the UCS satisf (4) for the input and the output. In the design of the VPLS, the can be defined as the designer wants, without considering the performance of the because its connection is usuall. So the latter condition is important to guarantee the stabilit b the VPLS. sstem input connection led limiter Fig. 2. UCS with Virtual Power Monitor Sstem Fig. 3. UCS with Virtual Power Limiter Sstem

3 D. Strateg to Guarantee Stabilit b VPLS Phsicall, the situation that (4) is not satisfied means the UCS generates energ and excessive is flowing out from the UCS. The UCS can be regulated to satisf (4) b the following methods: 1) decreasing the internal generated energ, 2) increasing the internal dissipated energ. In order to realize the first method, a solution is to regulate the input from the user s which is generating excessive energ. In order to realize the second method, a solution is to add some tunable energ dissipative elements in the UCS. The limiter applies some of the abovementioned methods based on the ed and regulates the outflow of the from the UCS. As shown in III-C, the asmptotic stabilit of the UCS with the VPLS is guaranteed if the UCS is tuned to satisf (4), regardless of whether the connection between the UCS and the VPLS is or phsical. IV. CASE STUDIES OF VPLS This section shows two examples of appling the VPLS to the robot sstems which cannot necessaril guarantee the stabilit due to unknown characteristics. A. Robot Control Sstem with Unknown Characteristics The first example of the UCS is a robot sstem. The details of the user s and the dnamics of the robot hardware as the user s are unknown for the designer of the VPLS. However, the joint displacement q R n and the joint velocit q R n are measurable b some sensors, and the input to the joint actuators from the user s u usr R n is lable. A structure of the VPLS for this robot sstem will be Fig.4. The driving force/torque u which is actuall applied to the robot hardware is u = W usr u usr + W csv + W v u v + u dis (8) u dis (t) R n is unknown input such as. u v (t) = K v (t) R n is a negative output feedback connected to the user s. K v R n n is a positive diagonal matrix of the feedback gain. W usr ( ), W csv ( ), W v ( ) R n n are positive diagonal matrices of varing weight functions, of which the limiter is comprised. The diagonal elements of W usr, W csv are greater than or equal to and less than or equal to 1, and W csv = I W usr. I R n n is the unit matrix. The following joint PD is chosen as the. = K p (q q d ) K d q (9) K p, K d R n n are positive diagonal matrices. q d R n is the target of the joint displacement. The target is the same as for the user s, as shown in the figure. If ou set the output = q, a conjugate pair is made b and. The stabilit evaluation (4) is equivalent to the following equation b (8). q T W usr (u usr ) dτ + q T u dis dτ E v + q T u dτ + q T W v K v q dτ (1) B. Man-Machine Sstem Next, the sstem which has some phsical interaction between human and robot is considered. This sstem is called a man-machine sstem. Generall, the stabilit of the man-machine sstem seems difficult to guarantee because the sstem includes the human dnamics as its unknown characteristics. A structure of the VPLS for the man-machine sstem will be Fig.5. For the man-machine sstem, the human is considered as the sstem. = u h (11) sstem sstem q d u usr u dis W usr u robot hardware u usr W usr u dis u robot hardware W v K W csv u v v W v K u v v limiter limiter q d human operator u h Fig. 4. Unknown Robot Control Sstem with Virtual Power Limiter Sstem Fig. 5. Power Assist Sstem with Virtual Power Limiter Sstem

4 u h is the force/torque which the human applies to the robot hardware phsicall. So the becomes the real here. The driving force/torque u which is actuall applied to the robot hardware is u = W usr u usr + u h + W v u v + u dis (12) The stabilit evaluation (4) is equivalent to the following equation b (12). C. Discussion q T W usr u usr dτ + q T u dis dτ E v + q T u dτ + q T W v K v q dτ (13) As mentioned in III-C, (4) is the sufficient condition to guarantee the asmptotic stabilit of the UCS with the VPLS. If the effect of the user s q T u usr, the effect of the robot hardware q T u, and the effect of the q T u dis are suppressed b the VPLS, the stabilit is guaranteed. Equation (1) and (13) are equivalent to (4). Even in the case that the ed is going out of (4), ou can tune the UCS to satisf (4) b changing W usr smaller and W v larger, in both cases of the unknown robot sstems and the manmachine sstems. This is equivalent to realize the automatic gain tuning b VPLS not to make the UCS unstable. Here the conditions to satisf (1) or (13) b the VPLS are discussed on the implementation level. 1) The values of and are reliable. 2) The calculations of W usr ( ), W csv ( ), W v ( ), and (4) in the /limiter are reliable. 3) In order to make W usr small enough, there is an access channel to the robot input u usr. The minimum implementation is that the actuators can be turned off b the VPLS. 4) In order to make W v large enough, there is an access channel to the dissipated energ. The minimum implementation is that the robot is equipped with the brakes hard enough to satisf (1) or (13). Equation (4) is satisfied and the asmptotic stabilit of the UCS with the VPLS is guaranteed on the above conditions. Note that there are no assumptions of the details of the user s, the user s, and the except their values q T u usr, q T u, and q T u dis are bounded. Even the passivit of the robot hardware does not need to be assumed. q T u dτ γ 2 V. DIFFERENCES FROM SIMILAR STRATEGIES (14) The similar sstems which guarantee the stabilit b ing the energ balance of a specific sstem have been alread proposed as the energ balance [3] and the time domain passivit observer/passivit ler [4], [5], [6]. Fig.6 shows the concept of these similar strategies. These guarantee the stabilit of the UCS b ing the energ balance and regulating the user s strategies apparentl passive, based on the assumption of the passivit of the user s. Namel, the stabilit cannot be guaranteed if the user s (with the ) is not passive. Although robot hardwares are originall passive, actual robot sstems are not necessaril passive due to low-level ler or. Fig.7 shows the concept of the VPLS. Unlike the similar strategies, the VPLS s not onl a part but also the whole of the UCS including the and. This is realized b adding the and ing the. The VPLS guarantees stabilit, even if actual robot sstems are not passive due to low-level ler, the, or other factors. There is another advantage of using two or more strategies. The user s can be aggressive onl for the performance. That is, if the performance is improved, the nonpassive is also permitted. Since the does not participate in actual if its connection is, it is not necessar to consider the performance in the design of the. On the other hand, the other similar sstems need to guarantee the passivit of the user s itself. Therefore onl the performance can be realized. VI. CONCLUSION In this paper, we redefined the limiter sstem and described the theoretical analsis. The VPLS is a new sstem led led Fig. 6. Concept of Other Energ Monitoring Sstems Fig. 7. Concept of Virtual Power Limiter Sstem

5 Fig. 8. Setup of Pneumatic Servo Sstem Fig. 9. Setup of Human Power Amplifier Sstem framework to guarantee unknown sstems. It was alread shown on the flexible manipulator [1] that the VPLS can stabilize the real robotic sstems. Now further experimental verification is performed on the pneumatic servo sstem of Fig.8 and the human amplifier sstem of Fig.9. In the preliminar experiments, the VPLS realizes the good stabilit of these sstems. ACKNOWLEDGMENT In the discussion to decide the name VPLS, Dr. Ruta Ozawa plaed an important role and made a lot of helpful suggestions. The author would like to appreciate his support. REFERENCES [1] K. Kanaoka and T. Yoshikawa: Passivit Monitor and Software Limiter which Guarantee Asmptotic Stabilit of Robot Control Sstems, Proc. IEEE ICRA 23, pp , 23. [2] S. Arimoto: Control Theor of Non-linear Mechanical Sstems A Passivit-based and Circuit-theoretic Approach. Oxford Universit Press, [3] Y. Yokokohji, T. Imaida, and T. Yoshikawa: Bilateral Control with Energ Balance Monitoring under Time-Varing Communication Dela, Proc. IEEE ICRA 2, pp , 2. [4] J.H. Ru, D.S. Kwon, and B. Hannaford: Stable Teleoperation with Time Domain Passivit Control, Proc. IEEE ICRA 22, pp , 22. [5] J.H. Ru, D.S. Kwon, and B. Hannaford: Control of a Flexible Manipulator with Noncollocated Feedback: Time Domain Passivit Approach, Control Problems in Robotics, Springer Tracts in Advanced Robotics, vol. 4, pp , 23. [6] J.H. Ru, Y.S. Kim, and B. Hannaford: Sampled and Continuous Time Passivit and Stabilit of Virtual Environments, Proc. IEEE ICRA 23, pp , 23.