An Iteration-Domain Filter for Controlling Transient Growth in Iterative Learning Control
|
|
- Melvyn Lane
- 5 years ago
- Views:
Transcription
1 21 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July 2, 21 WeC14.1 An Iteration-Domain Filter for Controlling Transient Growth in Iterative Learning Control Qing Liu and Douglas A. Bristow Abstract Transient growth is a problem in Iterative Learning Control (ILC) in which the tracking error temporarily grows very large during the learning process, before converging to a small value. While some ILC algorithms can guarantee monotonic convergence, there are limitations when the model is uncertain. This paper presents a new algorithm to reduce the transient growth in ILC. An domain filter, which can be applied to any linear ILC system, is proposed. The filter slows the learning process, in a controlled manner, to limit transient growth. Fundamental results relating the learning process convergence rate to explicit bounds on the transient growth are presented. Two examples that demonstrate the effectiveness of the method are presented: one in SISO design and one in network design. I I. INTRODUCTION terative learning control (ILC) is an approach to improve the tracking performance of a system that operates repetitively [1, 2]. The basic idea of the learning process is to use the control and error signal in the previous (s) to generate control signal for current system. Since it was first formulated in 1978 [3], ILC has been applied to a variety of applications. In these applications, model uncertainty at large frequencies requires some type of lowpass filtering to prevent transient growth. The lowpass filtering undesirably limits the learning bandwidth. Simple examples show that transient growth in exponentially stable ILC systems can mimic the response of an unstable system [1]. That is, the error can grow exponentially for several s before it begins to exponentially converge. Therefore, in application it is critical that the ILC be designed in such a way as to safely control transient growth. In previous works, transient growth is controlled using robust monotonically convergent algorithms like norm-optimal [4] and time domain method [5], but at the cost of asymptotic performance. Control of transient growth is even more challenging in networked systems, where centralized control methods such as norm Qing Liu is with the Department of Mechanical and Aerospace Engineering, Missouri University of Science and Technology, Rolla, MO 6549 USA Douglas A. Bristow is with the Department of Mechanical and Aerospace Engineering, Missouri University of Science and Technology, Rolla, MO 6549 USA (corresponding author, dbristow@mst.edu) optimal and frequency domain may not apply well. Therefore, other methods of controlling transient growth are necessary. In this paper, we will present an domain filter that can be applied to any linear ILC algorithm to control transient growth. The filter contains a tuning variable that provides a tradeoff between growth reduction and convergence rate. In section II, we summarize the transient growth problem in ILC. In section III, we present our domain filter and prove an explicit relationship between the filter s convergence rate and the transient growth. A tuning process is presented, which is useful in practice when the system is not well modeled or the system scale is large to easily analyze. In section IV, a P-type ILC system and a master-slave system are presented to demonstrate the effectiveness of the domain filter. Finally in section V, we describe the future works. II. ILC TRANSIENT GROWTH BACKGROUND A discrete-time, linear time invariant system has the form 1, (1) where, is the time during a given. Define 1, 2,,, and, 1,, 1, where means the th (operation) of learning. Let 1, 2,, be the vector of desired outputs. Using the lifted setting [6], the system can be written as, (2) where is the Toeplitz matrix [7] of the plant:, (3) where,,, is the impulse response of the system. There are a number of ILC algorithms that can be designed for the system (1). The reader is referred to [1,2] for a categorization and listing. Here, we take a general approach and assume that any linear, first-order ILC algorithm is used. Such an algorithm can always be written as, (4) where and are NxN learning gains, and the error. Assuming, we have that, and the closed-loop equation for the error can be written as 1. (5) /1/$ AACC 239
2 When, 1 1, where. The convergence dynamics can be obtained by writing (5) as. (6) Clearly the system is convergent if 1, for 1,,, where is the i th eigenvalue of. From equation (6),. Thus, we say that the system is monotonically convergent if 1 [1], because the error approaches the asymptotic solution monotonically. If the system is not monotonically convergent, there will be a transient growth, and a tight bound on the magnitude of the growth is difficult to calculate. While, it is well known that, where, is a (non unique) matrix of the eigenvectors of, and is the spectral radius of, or the largest eigenvalue of [8]. In the existing ILC design method, the convergence of system is not hard to achieve, but the transient growth is usually a problem. We will illustrate this in the following two examples. After developing our new approach in Section III, we return to these examples in Section IV to demonstrate its effectiveness. P-type ILC system performance Consider the P-type ILC algorithm [6], which is the simplest method, as an example. Then 1, where is a constant. From (6). (7) Clearly the system is exponentially convergent if 1 1. RMS error large transient growth convergence speed λ Figure 1: Large transient in a stable learning control system.. Here we use,.. 1,.6, 4. The eigenvalues are all located at.4, well inside the unit disk, but 4 orders of magnitude growth are observed in the transient. In fact, convergence of this system is quite robust. Using only the sign and magnitude of, we can pick a sufficiently small (or ) to provide robust convergence. Although convergence depends only on the eigenvalues, and thus only on and, the transient growth bound depends on the full system. Thus, as shown in figure 1, it is easy to construct an example where eigenvalues are small but growth is large. Although the system is convergent and robust to dynamic variations in P, it is not acceptable in practice due to the large transient growth. Master-slave networked systems In some applications such as UAV [9], robotic systems [1] and manufacturing [11], coordinated, or networked, systems are used. The master-slave networked system shown in figure 2 is a simple type of networked system. Here the first system tracks and system 1 tracks system. The connection among systems is decentralized, which is especially useful if the network contains a large number of systems. u 1 j u 1( j 1) e1( j 1 ) u 2 j P 1 P 2 e 1 j u 2( j 1) e2 ( j 1 ) e 2 j y d y 1j y 1 j Figure 2: A master-slave networked system y 2 j When the number of systems is large, it is difficult to use any existing ILC method to control. Particularly in the case of lifted-systems based approaches, it may be difficult to construct the nnxnn lifted-system. Here we assume that all systems are identical,, n is the number of systems in the group. Use same ILC learning law for each system,, and, and further assume that L u and L e are selected such that 1 where. By selecting L u and L e in this manner, each system is capable of monotonically tracking its input as a stand-alone system. However, as we will show, when the systems are interconnected, monotonicity is lost. 24
3 For the networked system we find that the error is given by, Thus, where (8) 1 1,,,,,,,,,,,,,,,,. Clearly, convergence of the networked system is inherited from the individual sub-systems, because the eigenvalues of are simply the eigenvalues of in the sub-systems. However, 1 does not imply 1. Therefore, monotonicity of the networked system is not inherited from the individual sub-systems. Developing a method that extends monotonicity of the individual systems to monotonicity of the networked system is a primary motivator for this work. Remark: In the above example, it may be possible to leverage the Toeplitz structure of to design decentralized controllers and. However, a simple extension of the above to the case where each system is unique would increase the complexity of such an approach. From the above examples, it is not hard to see that transient growth of convergent systems is a big problem. Our approach can solve the transient problem without changing final performance by slowing the system down. III. ITERATION DOMAIN FILTER Assume we are given the learning gains and, which can be designed by any ILC algorithm, like P-type, PD-type, time-varying method and norm-optimal design. Assume also that the ILC system is known to be stable ( 1) but not monotonically convergent ( 1). Define the scalar sequence as a monotonically increasing filter with the following properties, lim 1. (1) Consider the modified learning algorithm where the prefilter is added to (4) as,, (11) (9) Our modified learning algorithm can be interpreted as a forgetting-factor algorithm, where the forgetting factor is aproaches zero with increasing. The closed-loop error with the modified learning algorithm can be written as 1 1, (12) where,. For every,,1,2, we define the f j - steady-state error as, 1. (13) Note that always exists because the eigenvalues of are smaller than 1. The f j -steady-state error is the error that would be obtained if f j were frozen for all future s. That is, is the steady-state error when. For the -varying filter, the error contains two parts, the -steady-state error and the transient error in learning process, as shown in figure 3. The transient error is the source of transient growth problem described in section II. To isolate the two parts, the transient error is defined in a special way. (14) RM Steady state error System error Transient error Figure 3: System error, steady-state error and transient error We can derive the propagation of as follows, 1 1, (15) where. The -steady-state error is a smooth and bounded function, so there exists an L such that, where is the Lipschitz constant of [12]. 241
4 Define as the maximum rate of increase of as. The main contribution of this work is the following theorem, which proves that the transient growth is proportional to, the rate of increase of. Thus, the transient error can be controlled with appropriate selection of. Theorem: Given the system (2), the modified ILC algorithm (11). If 1 and, then there exists a learning rate,1, for all, such that the transient error is bounded by, for all j. Proof: From (15), there is. Since 1, there is. Furthermore, since 1, so there exists K> and 1 such that. Using the Lipschitz bound on,. Then,. Thus, the transient error bound is proportional to. It remains to show that there exists an,1 to satisfy the bound. Choosing such that. (16) Then,, which completes the proof. Remark: From the above proof, it is clear that when P is known one can explicitly calculate to achieve the desired bound on transient growth. However, if the system is not well known or has a large size (for example, the large scale coordinated systems), calculation of an appropriate calculate is challenging. For such systems, a tuning process is always practical. Because always exists for convergent systems, it is always possible to tune through a trial and error process to achieve the desired behavior. Remark: It is notable from (equation 16) that is inversely proportional to K. Since, K is a measure of the transient growth of the system. Therefore, the larger the system s transient growth, the smaller one must choose, and thus the slower the convergence. Clearly, gives the fastest rate of increase for, of course, should not exceed one, so we use min,1. When, 1, the steady state error 1 1, which is same as the final system error without the filter from equation (5). The final performance will not be influenced by the filter. From the analysis above, we can see that the modified learning algorithm (11) will decrease the transient growth, while maintaining the same asymptotic performance as the unmodified learning algorithm (4). The tradeoff is slower convergence. IV. SIMULATED RESULTS In this part, we will present two examples to demonstrate the effectiveness of the domain filter. Example 1: SISO Tracking First we use a second order discrete time system [5]. (17).. The reference is given in figure 4. y d t Figure 4: reference To demonstrate our filter, we use a P-type ILC design, which is known to result in transient growth. While other ILC designs such as norm-optimal or model inverse will result in monotonic convergence, we can treat this example as a highly uncertain system, such that those approaches will not be benefitial. Let 1,.6, then.6. This is the same system we examined in Section II. The system is convergent since.4, but it is not monotonically convergent since We use the filter. In this case, we determine that 1, 2,.4, so according to (16) there is 31. Clearly this value of is impractically slow. However, (16) is a conservative condition, as are estimates of K. We find that the tuning process will be more practical in this case. Several choices of, 1,.1,.1, are shown in figure 5. Note that algorithm (11) with 1 is identical to algorithm (4), and thus this is the nominal P-type ILC. The results are shown in table 1 and figure 6. Table 1: Transient error with different Transient error
5 f j RMS (transient error) 1.5 α=.1 α= Figure 5: Filter α=.1 α= Figure 6: for ILC system in example 1, with filter, when is smaller, the transient error is smaller, and the convergent speed is slower. As expected, we see a large transient growth for 1, with decreasing growth, but slower convergence as is decreased. To further reduce transient growth much smaller values of are needed. Alternatively, we consider other forms of the filter that also satisfy. Beginning with the results in figure 6, we see that the transient error appears to grow suddenly when is close to, or at, one. Therefore, rather than continuing to select smaller values of, in the algorithm, it may be more advantageous to select a algorithm with a slower transition to 1. Therefore an exponential filter 11 is used. Here the largest increase rate in occurs from j= to j=1, so. Several choices of, 1,.2,.1, are selected and the filter and simulation results are shown in table 2, figure 7and figure 8. Clearly the exponential filter is more effective at reducing the transient growth. Although it is slower than the linear filter, as evident by the asymptotic convergence rates, the improved reduction in transient growth at early s makes it overall the more practical option. Table 2: Transient error with exponential filter Transient error f j RMS(transient error) 1.5 α=.2 α= Figure 7: Filter Figure 8: for ILC system in example 1, with filter 1 1, when is smaller, the transient error is smaller, and the convergent speed is slower. These results above shows that the proposed varying filter reduces transient error, while the trade off is slower convergent speed. Furthermore, the tuning approach appears to work quite well. Clearly, however, the best -varying filter is not necessarily the fastest one. Example 2: Master-slave Networked System Next we examine the master-slave networked system described in Section II. Let α=.2 α=.1..., (18) and use PD-type ILC [1] for each system, 1, (19) The master system tracks the reference in Figure 4. According to (9), the convergence is provided by.251. For each individual system, there is.6882, which indicates the monotonic convergence of each system. But 1 cannot be ensured even with 1, because will be changed every time when a new slave system is attached. Thus the system is convergent but not necessarily monotonically convergent. In Figure 9 we show that the networked system is not monotonic for the nominal learning algorithm (4). In fact, the transient growth increases with each additional system. The largest growth in the networked system is 1, which 243
6 appears on the 2 th system. Clearly adding more systems will result in increased growth. RMS error Figure 9: Transient growth for the networked system using the nominal learning algorithm Since it shows that the exponential filter works better in Example 1, we apply an exponential filter 11 on the system. Using the modified learning algorithm, (11), we tune until the transient growth is effectively removed. Figure 1 shows the series of the error with.1. RMS error increasing transient growth with increasing # of systems master 5th sys 1th sys 15th sys 2th sys blue: master sys green: 1th sys red: 2th sys α= Figure 1: Transient growth for the networked system using the modified learning algorithm V. CONCLUSION In this paper, our work is focused on transient growth problem in ILC system(s). First we reviewed the transient growth problem, and showed that although the convergence of the ILC system is not hard to provide with existing algorithms, when the system model is uncertain or too complicated such as in a network system, a suitable transient growth bounded is challenging. We proposed an domain filter to control transient error. Analysis shows that such a filter can always be found to bound the transient error to low magnitude. With the filter applied on a stable ILC system designed by any algorithm, it is possible to trade convergent speed for lower transient growth. Two examples were used to demonstrate the utility of this method in SISO design and in a master-slave network design. REFERENCES [1] D. Bristow, M. Tharayil and A. Alleyne, A Servey of Iterative Learning Control, IEEE Control Systems Magazine, vol. 26, no. 3, 26, pp [2] H. Ahn, Y. Chen and K. Moore, Iterative Learning Control: Brief Survey and Categorization, IEEE Transactions on Systems, Man and Cybernetics Part C: Applications and Reviews, v 37, n 6, Nov. 27, p [3] M. Uchiyama, Formation of high-speed motion pattern of a mechanical arm by trial, Trans. Soc. Instrument Contr. Engineers, vol. 14, no. 6, pp , [4] N.Amann, D.H.Owens and E. Rogers, Iterative learning control for discrete-time systemswith exponential rate of convergence, IEEE Proc-Control Theory Appl., Vol. 143, No. 2, March 1996 [5] K. Moore, Y. Chen, and V. Bahl, Monotonically convergent iterative learning control for linear discrete-time systems, Automatica, vol. 41, issue 9, Sep. 25, pp [6] M. Phan and R. Longman, A mathematical theory of learning control for linear discrete multivariable systems, in Proc. of the AIAA/AAS Astrodynamics Specialist Conference, 1988 [7] T. Kailath, Linear Systems. Englewood Cliffs, NJ : Prentice-Hall, 198. [8] L. N. Trefethen and M. Embree, Spectra And Pseudospectra. Princeton, NJ: Princeton University Press, 25 [9] J. How, B. Bethke, A. Frank, D. Dale and J. Vian, Real-Time Indoor Autonomous Vehicle Test Enviroment, IEEE Control Systems Magazine, vol. 28, no. 2, Apr. 28, pp [1] C. Belta and V. Kumar, Abstract and Control for Groups of Robots, IEEE Trans. Robotics, vol. 2, no. 5, 24, pp [11] K. Yoram, Cross-coupled Biaxial Computer Control for Manufacturing Systems, Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME, v 12, n 4, Dec. 198, p [12] Hassan K. Khalil, Nonlinear Systems, Upper Saddle River, NJ: Prentice Hall, 1996 Just as with the nominal algorithm, adding more systems will result in the appearance, and increased size, of transient growth. However, with the modified algorithm it is a simple process of re-tuning the parameter to achieve the desired transient behavior. 244
Iterative Learning Control Analysis and Design I
Iterative Learning Control Analysis and Design I Electronics and Computer Science University of Southampton Southampton, SO17 1BJ, UK etar@ecs.soton.ac.uk http://www.ecs.soton.ac.uk/ Contents Basics Representations
More informationPERIODIC signals are commonly experienced in industrial
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 2, MARCH 2007 369 Repetitive Learning Control of Nonlinear Continuous-Time Systems Using Quasi-Sliding Mode Xiao-Dong Li, Tommy W. S. Chow,
More informationA Discrete Robust Adaptive Iterative Learning Control for a Class of Nonlinear Systems with Unknown Control Direction
Proceedings of the International MultiConference of Engineers and Computer Scientists 16 Vol I, IMECS 16, March 16-18, 16, Hong Kong A Discrete Robust Adaptive Iterative Learning Control for a Class of
More informationRiccati difference equations to non linear extended Kalman filter constraints
International Journal of Scientific & Engineering Research Volume 3, Issue 12, December-2012 1 Riccati difference equations to non linear extended Kalman filter constraints Abstract Elizabeth.S 1 & Jothilakshmi.R
More informationOptimization based robust control
Optimization based robust control Didier Henrion 1,2 Draft of March 27, 2014 Prepared for possible inclusion into The Encyclopedia of Systems and Control edited by John Baillieul and Tariq Samad and published
More informationPerformance of an Adaptive Algorithm for Sinusoidal Disturbance Rejection in High Noise
Performance of an Adaptive Algorithm for Sinusoidal Disturbance Rejection in High Noise MarcBodson Department of Electrical Engineering University of Utah Salt Lake City, UT 842, U.S.A. (8) 58 859 bodson@ee.utah.edu
More informationarxiv: v2 [cs.ro] 26 Sep 2016
Distributed Iterative Learning Control for a Team of Quadrotors Andreas Hock and Angela P Schoellig arxiv:1635933v [csro] 6 Sep 16 Abstract The goal of this work is to enable a team of quadrotors to learn
More informationThe Rationale for Second Level Adaptation
The Rationale for Second Level Adaptation Kumpati S. Narendra, Yu Wang and Wei Chen Center for Systems Science, Yale University arxiv:1510.04989v1 [cs.sy] 16 Oct 2015 Abstract Recently, a new approach
More informationPart 1: Introduction to the Algebraic Approach to ILC
IEEE ICMA 2006 Tutorial Workshop: Control Algebraic Analysis and Optimal Design Presenters: Contributor: Kevin L. Moore Colorado School of Mines YangQuan Chen Utah State University Hyo-Sung Ahn ETRI, Korea
More informationControl Systems Theory and Applications for Linear Repetitive Processes
Eric Rogers, Krzysztof Galkowski, David H. Owens Control Systems Theory and Applications for Linear Repetitive Processes Springer Contents 1 Examples and Representations 1 1.1 Examples and Control Problems
More informationSimple Learning Control Made Practical by Zero-Phase Filtering: Applications to Robotics
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL 49, NO 6, JUNE 2002 753 Simple Learning Control Made Practical by Zero-Phase Filtering: Applications to Robotics Haluk
More informationPredictive Iterative Learning Control using Laguerre Functions
Milano (Italy) August 28 - September 2, 211 Predictive Iterative Learning Control using Laguerre Functions Liuping Wang Eric Rogers School of Electrical and Computer Engineering, RMIT University, Victoria
More informationNonlinear Tracking Control of Underactuated Surface Vessel
American Control Conference June -. Portland OR USA FrB. Nonlinear Tracking Control of Underactuated Surface Vessel Wenjie Dong and Yi Guo Abstract We consider in this paper the tracking control problem
More informationSystem Identification Using a Retrospective Correction Filter for Adaptive Feedback Model Updating
9 American Control Conference Hyatt Regency Riverfront, St Louis, MO, USA June 1-1, 9 FrA13 System Identification Using a Retrospective Correction Filter for Adaptive Feedback Model Updating M A Santillo
More informationMANY adaptive control methods rely on parameter estimation
610 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 52, NO 4, APRIL 2007 Direct Adaptive Dynamic Compensation for Minimum Phase Systems With Unknown Relative Degree Jesse B Hoagg and Dennis S Bernstein Abstract
More informationCONTROL SYSTEMS, ROBOTICS AND AUTOMATION - Vol. XII - Lyapunov Stability - Hassan K. Khalil
LYAPUNO STABILITY Hassan K. Khalil Department of Electrical and Computer Enigneering, Michigan State University, USA. Keywords: Asymptotic stability, Autonomous systems, Exponential stability, Global asymptotic
More informationDesign strategies for iterative learning control based on optimal control
Selected Topics in Signals, Systems and Control Vol. 2, September 2 Design strategies for iterative learning control based on optimal control Rob Tousain, Eduard van der Meché and Okko Bosgra Mechanical
More informationAdaptive Robust Tracking Control of Robot Manipulators in the Task-space under Uncertainties
Australian Journal of Basic and Applied Sciences, 3(1): 308-322, 2009 ISSN 1991-8178 Adaptive Robust Tracking Control of Robot Manipulators in the Task-space under Uncertainties M.R.Soltanpour, M.M.Fateh
More information458 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 16, NO. 3, MAY 2008
458 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL 16, NO 3, MAY 2008 Brief Papers Adaptive Control for Nonlinearly Parameterized Uncertainties in Robot Manipulators N V Q Hung, Member, IEEE, H D
More informationIterative learning control (ILC) is based on the notion
Iterative learning control (ILC) is based on the notion that the performance of a system that executes the same task multiple times can be improved by learning from previous executions (trials, iterations,
More informationDecoupled Feedforward Control for an Air-Conditioning and Refrigeration System
American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July, FrB1.4 Decoupled Feedforward Control for an Air-Conditioning and Refrigeration System Neera Jain, Member, IEEE, Richard
More informationH-infinity Model Reference Controller Design for Magnetic Levitation System
H.I. Ali Control and Systems Engineering Department, University of Technology Baghdad, Iraq 6043@uotechnology.edu.iq H-infinity Model Reference Controller Design for Magnetic Levitation System Abstract-
More informationIterative Methods for Eigenvalues of Symmetric Matrices as Fixed Point Theorems
Iterative Methods for Eigenvalues of Symmetric Matrices as Fixed Point Theorems Student: Amanda Schaeffer Sponsor: Wilfred M. Greenlee December 6, 007. The Power Method and the Contraction Mapping Theorem
More informationDesign and Stability Analysis of Single-Input Fuzzy Logic Controller
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL. 30, NO. 2, APRIL 2000 303 Design and Stability Analysis of Single-Input Fuzzy Logic Controller Byung-Jae Choi, Seong-Woo Kwak,
More informationOn the History, Accomplishments, and Future of the Iterative Learning Control Paradigm
Control Paradigm On the History, Accomplishments, and Future of the Control Paradigm Presenter: Contributors: Kevin L. Moore G.A. Dobelman Distinguished Chair and Professor of Engineering Division of Engineering
More information1348 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL. 34, NO. 3, JUNE 2004
1348 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL 34, NO 3, JUNE 2004 Direct Adaptive Iterative Learning Control of Nonlinear Systems Using an Output-Recurrent Fuzzy Neural
More informationANALYSIS AND SYNTHESIS OF DISTURBANCE OBSERVER AS AN ADD-ON ROBUST CONTROLLER
ANALYSIS AND SYNTHESIS OF DISTURBANCE OBSERVER AS AN ADD-ON ROBUST CONTROLLER Hyungbo Shim (School of Electrical Engineering, Seoul National University, Korea) in collaboration with Juhoon Back, Nam Hoon
More informationTHIS paper deals with robust control in the setup associated
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 10, OCTOBER 2005 1501 Control-Oriented Model Validation and Errors Quantification in the `1 Setup V F Sokolov Abstract A priori information required for
More informationAADECA 2006 XXº Congreso Argentino de Control Automático ITERATIVE LEARNING CONTROL APPLIED TO NONLINEAR BATCH REACTOR E.J.
ITERATIVE LEARNING CONTROL APPLIED TO NONLINEAR BATCH REACTOR E.J. ADAM (1) Institute of Technological Development for the Chemical Industry (INTEC), CONICET Universidad Nacional del Litoral (UNL). Güemes
More informationIN recent years, controller design for systems having complex
818 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL 29, NO 6, DECEMBER 1999 Adaptive Neural Network Control of Nonlinear Systems by State and Output Feedback S S Ge, Member,
More informationA 2D Systems Approach to Iterative Learning Control with Experimental Validation
Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 28 A 2D Systems Approach to Iterative Learning Control with Experimental Validation Lukasz
More informationL 1 Adaptive Controller for Multi Input Multi Output Systems in the Presence of Unmatched Disturbances
28 American Control Conference Westin Seattle Hotel, Seattle, Washington, USA June -3, 28 FrA4.4 L Adaptive Controller for Multi Input Multi Output Systems in the Presence of Unmatched Disturbances Chengyu
More informationNonlinear Adaptive Robust Control. Theory and Applications to the Integrated Design of Intelligent and Precision Mechatronic Systems.
A Short Course on Nonlinear Adaptive Robust Control Theory and Applications to the Integrated Design of Intelligent and Precision Mechatronic Systems Bin Yao Intelligent and Precision Control Laboratory
More informationGAIN SCHEDULING CONTROL WITH MULTI-LOOP PID FOR 2- DOF ARM ROBOT TRAJECTORY CONTROL
GAIN SCHEDULING CONTROL WITH MULTI-LOOP PID FOR 2- DOF ARM ROBOT TRAJECTORY CONTROL 1 KHALED M. HELAL, 2 MOSTAFA R.A. ATIA, 3 MOHAMED I. ABU EL-SEBAH 1, 2 Mechanical Engineering Department ARAB ACADEMY
More informationMultipredictive Adaptive Control of Arc Welding Trailing Centerline Temperature
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 8, NO. 1, JANUARY 2000 159 Multipredictive Adaptive Control of Arc Welding Trailing Centerline Temperature T. O. Santos, R. B. Caetano, J. M. Lemos,
More informationExponential Controller for Robot Manipulators
Exponential Controller for Robot Manipulators Fernando Reyes Benemérita Universidad Autónoma de Puebla Grupo de Robótica de la Facultad de Ciencias de la Electrónica Apartado Postal 542, Puebla 7200, México
More informationAUTOMATIC CONTROL COMMUNICATION SYSTEMS LINKÖPING
"!# $ %'&)(+* &-,.% /03254-687:9@?A?AB54 C DFEHG)IJ237KI#L BM>A>@ION B5P Q ER0EH?@EHBM4.B3PTSU;V68BMWX2368ERY@BMI Q 7K[25>@6AWX7\4)6]B3PT^_IH7\Y\6A>AEHYK25I#^_4`MER47K7\>AER4` a EH4GbN
More informationHere represents the impulse (or delta) function. is an diagonal matrix of intensities, and is an diagonal matrix of intensities.
19 KALMAN FILTER 19.1 Introduction In the previous section, we derived the linear quadratic regulator as an optimal solution for the fullstate feedback control problem. The inherent assumption was that
More informationDISTURBANCE OBSERVER BASED CONTROL: CONCEPTS, METHODS AND CHALLENGES
DISTURBANCE OBSERVER BASED CONTROL: CONCEPTS, METHODS AND CHALLENGES Wen-Hua Chen Professor in Autonomous Vehicles Department of Aeronautical and Automotive Engineering Loughborough University 1 Outline
More informationWE EXAMINE the problem of controlling a fixed linear
596 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 43, NO 5, MAY 1998 Controller Switching Based on Output Prediction Errors Judith Hocherman-Frommer, Member, IEEE, Sanjeev R Kulkarni, Senior Member, IEEE,
More informationDesign Artificial Nonlinear Controller Based on Computed Torque like Controller with Tunable Gain
World Applied Sciences Journal 14 (9): 1306-1312, 2011 ISSN 1818-4952 IDOSI Publications, 2011 Design Artificial Nonlinear Controller Based on Computed Torque like Controller with Tunable Gain Samira Soltani
More informationPerforming Aggressive Maneuvers using Iterative Learning Control
2009 IEEE International Conference on Robotics and Automation Kobe International Conference Center Kobe, Japan, May 12-17, 2009 Performing Aggressive Maneuvers using Iterative Learning Control Oliver Purwin
More informationStability Analysis of a Proportional with Intermittent Integral Control System
American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July, ThB4. Stability Analysis of a Proportional with Intermittent Integral Control System Jin Lu and Lyndon J. Brown Abstract
More informationCANONICAL LOSSLESS STATE-SPACE SYSTEMS: STAIRCASE FORMS AND THE SCHUR ALGORITHM
CANONICAL LOSSLESS STATE-SPACE SYSTEMS: STAIRCASE FORMS AND THE SCHUR ALGORITHM Ralf L.M. Peeters Bernard Hanzon Martine Olivi Dept. Mathematics, Universiteit Maastricht, P.O. Box 616, 6200 MD Maastricht,
More informationA NONLINEAR TRANSFORMATION APPROACH TO GLOBAL ADAPTIVE OUTPUT FEEDBACK CONTROL OF 3RD-ORDER UNCERTAIN NONLINEAR SYSTEMS
Copyright 00 IFAC 15th Triennial World Congress, Barcelona, Spain A NONLINEAR TRANSFORMATION APPROACH TO GLOBAL ADAPTIVE OUTPUT FEEDBACK CONTROL OF RD-ORDER UNCERTAIN NONLINEAR SYSTEMS Choon-Ki Ahn, Beom-Soo
More informationAPPLICATION OF D-K ITERATION TECHNIQUE BASED ON H ROBUST CONTROL THEORY FOR POWER SYSTEM STABILIZER DESIGN
APPLICATION OF D-K ITERATION TECHNIQUE BASED ON H ROBUST CONTROL THEORY FOR POWER SYSTEM STABILIZER DESIGN Amitava Sil 1 and S Paul 2 1 Department of Electrical & Electronics Engineering, Neotia Institute
More informationRobust Control. 2nd class. Spring, 2018 Instructor: Prof. Masayuki Fujita (S5-303B) Tue., 17th April, 2018, 10:45~12:15, S423 Lecture Room
Robust Control Spring, 2018 Instructor: Prof. Masayuki Fujita (S5-303B) 2nd class Tue., 17th April, 2018, 10:45~12:15, S423 Lecture Room 2. Nominal Performance 2.1 Weighted Sensitivity [SP05, Sec. 2.8,
More informationA Robust Controller for Scalar Autonomous Optimal Control Problems
A Robust Controller for Scalar Autonomous Optimal Control Problems S. H. Lam 1 Department of Mechanical and Aerospace Engineering Princeton University, Princeton, NJ 08544 lam@princeton.edu Abstract Is
More informationInitial condition issues on iterative learning control for non-linear systems with time delay
Internationa l Journal of Systems Science, 1, volume, number 11, pages 15 ±175 Initial condition issues on iterative learning control for non-linear systems with time delay Mingxuan Sun and Danwei Wang*
More informationH 2 Adaptive Control. Tansel Yucelen, Anthony J. Calise, and Rajeev Chandramohan. WeA03.4
1 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July, 1 WeA3. H Adaptive Control Tansel Yucelen, Anthony J. Calise, and Rajeev Chandramohan Abstract Model reference adaptive
More informationDESIGN OF ROBUST CONTROL SYSTEM FOR THE PMS MOTOR
Journal of ELECTRICAL ENGINEERING, VOL 58, NO 6, 2007, 326 333 DESIGN OF ROBUST CONTROL SYSTEM FOR THE PMS MOTOR Ahmed Azaiz Youcef Ramdani Abdelkader Meroufel The field orientation control (FOC) consists
More informationIteration-Domain Robust Iterative Learning Control
Iteration-Domain Robust Control Iteration-Domain Robust Control Presenter: Contributors: Kevin L. Moore G.A. Dobelman Distinguished Chair and Professor of Engineering Division of Engineering Colorado School
More informationOn the Scalability in Cooperative Control. Zhongkui Li. Peking University
On the Scalability in Cooperative Control Zhongkui Li Email: zhongkli@pku.edu.cn Peking University June 25, 2016 Zhongkui Li (PKU) Scalability June 25, 2016 1 / 28 Background Cooperative control is to
More information554 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 2, FEBRUARY and such that /$ IEEE
554 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 2, FEBRUARY 2010 REFERENCES [1] M. Fliess, J. Levine, P. Martin, and P. Rouchon, Flatness and defect of nonlinear systems: Introductory theory and
More informationBackstepping Control of Linear Time-Varying Systems With Known and Unknown Parameters
1908 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 11, NOVEMBER 2003 Backstepping Control of Linear Time-Varying Systems With Known and Unknown Parameters Youping Zhang, Member, IEEE, Barış Fidan,
More informationLifted approach to ILC/Repetitive Control
Lifted approach to ILC/Repetitive Control Okko H. Bosgra Maarten Steinbuch TUD Delft Centre for Systems and Control TU/e Control System Technology Dutch Institute of Systems and Control DISC winter semester
More informationLearning Model Predictive Control for Iterative Tasks: A Computationally Efficient Approach for Linear System
Learning Model Predictive Control for Iterative Tasks: A Computationally Efficient Approach for Linear System Ugo Rosolia Francesco Borrelli University of California at Berkeley, Berkeley, CA 94701, USA
More informationAdaptive Dynamic Inversion Control of a Linear Scalar Plant with Constrained Control Inputs
5 American Control Conference June 8-, 5. Portland, OR, USA ThA. Adaptive Dynamic Inversion Control of a Linear Scalar Plant with Constrained Control Inputs Monish D. Tandale and John Valasek Abstract
More informationRobust Stabilization of Non-Minimum Phase Nonlinear Systems Using Extended High Gain Observers
28 American Control Conference Westin Seattle Hotel, Seattle, Washington, USA June 11-13, 28 WeC15.1 Robust Stabilization of Non-Minimum Phase Nonlinear Systems Using Extended High Gain Observers Shahid
More informationA Sliding Mode Controller Using Neural Networks for Robot Manipulator
ESANN'4 proceedings - European Symposium on Artificial Neural Networks Bruges (Belgium), 8-3 April 4, d-side publi., ISBN -9337-4-8, pp. 93-98 A Sliding Mode Controller Using Neural Networks for Robot
More informationDistributed Adaptive Consensus Protocol with Decaying Gains on Directed Graphs
Distributed Adaptive Consensus Protocol with Decaying Gains on Directed Graphs Štefan Knotek, Kristian Hengster-Movric and Michael Šebek Department of Control Engineering, Czech Technical University, Prague,
More informationTHE DESIGN OF ACTIVE CONTROLLER FOR THE OUTPUT REGULATION OF LIU-LIU-LIU-SU CHAOTIC SYSTEM
THE DESIGN OF ACTIVE CONTROLLER FOR THE OUTPUT REGULATION OF LIU-LIU-LIU-SU CHAOTIC SYSTEM Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University
More informationAdvanced Aerospace Control. Marco Lovera Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano
Advanced Aerospace Control Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano ICT for control systems engineering School of Industrial and Information Engineering Aeronautical Engineering
More informationLecture 13: Internal Model Principle and Repetitive Control
ME 233, UC Berkeley, Spring 2014 Xu Chen Lecture 13: Internal Model Principle and Repetitive Control Big picture review of integral control in PID design example: 0 Es) C s) Ds) + + P s) Y s) where P s)
More informationDATA receivers for digital transmission and storage systems
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: EXPRESS BRIEFS, VOL. 52, NO. 10, OCTOBER 2005 621 Effect of Loop Delay on Phase Margin of First-Order Second-Order Control Loops Jan W. M. Bergmans, Senior
More informationApplied Nonlinear Control
Applied Nonlinear Control JEAN-JACQUES E. SLOTINE Massachusetts Institute of Technology WEIPING LI Massachusetts Institute of Technology Pearson Education Prentice Hall International Inc. Upper Saddle
More informationOUTPUT REGULATION OF RÖSSLER PROTOTYPE-4 CHAOTIC SYSTEM BY STATE FEEDBACK CONTROL
International Journal in Foundations of Computer Science & Technology (IJFCST),Vol., No., March 01 OUTPUT REGULATION OF RÖSSLER PROTOTYPE-4 CHAOTIC SYSTEM BY STATE FEEDBACK CONTROL Sundarapandian Vaidyanathan
More informationRobust control of resistive wall modes using pseudospectra
Robust control of resistive wall modes using pseudospectra M. Sempf, P. Merkel, E. Strumberger, C. Tichmann, and S. Günter Max-Planck-Institut für Plasmaphysik, EURATOM Association, Garching, Germany GOTiT
More informationVideo 6.1 Vijay Kumar and Ani Hsieh
Video 6.1 Vijay Kumar and Ani Hsieh Robo3x-1.6 1 In General Disturbance Input + - Input Controller + + System Output Robo3x-1.6 2 Learning Objectives for this Week State Space Notation Modeling in the
More informationDesign of Norm-Optimal Iterative Learning Controllers: The Effect of an Iteration-Domain Kalman Filter for Disturbance Estimation
Design of Norm-Optimal Iterative Learning Controllers: The Effect of an Iteration-Domain Kalman Filter for Disturbance Estimation Nicolas Degen, Autonomous System Lab, ETH Zürich Angela P. Schoellig, University
More informationNew Recursive-Least-Squares Algorithms for Nonlinear Active Control of Sound and Vibration Using Neural Networks
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 12, NO. 1, JANUARY 2001 135 New Recursive-Least-Squares Algorithms for Nonlinear Active Control of Sound and Vibration Using Neural Networks Martin Bouchard,
More informationADAPTIVE control of uncertain time-varying plants is a
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 1, JANUARY 2011 27 Supervisory Control of Uncertain Linear Time-Varying Systems Linh Vu, Member, IEEE, Daniel Liberzon, Senior Member, IEEE Abstract
More informationSteady-state DKF. M. Sami Fadali Professor EE UNR
Steady-state DKF M. Sami Fadali Professor EE UNR 1 Outline Stability of linear estimators. Lyapunov equation. Uniform exponential stability. Steady-state behavior of Lyapunov equation. Riccati equation.
More informationOn the Current Error Based Sampled-data Iterative Learning Control with Reduced Memory Capacity
International Journal of Automation and Computing 12(3), June 2015, 307-315 DOI: 101007/s11633-015-0890-1 On the Current Error Based Sampled-data Iterative Learning Control with Reduced Memory Capacity
More informationAn Optimal Tracking Approach to Formation Control of Nonlinear Multi-Agent Systems
AIAA Guidance, Navigation, and Control Conference 13-16 August 212, Minneapolis, Minnesota AIAA 212-4694 An Optimal Tracking Approach to Formation Control of Nonlinear Multi-Agent Systems Ali Heydari 1
More informationLPV Decoupling and Input Shaping for Control of Diesel Engines
American Control Conference Marriott Waterfront, Baltimore, MD, USA June -July, WeB9.6 LPV Decoupling and Input Shaping for Control of Diesel Engines Javad Mohammadpour, Karolos Grigoriadis, Matthew Franchek,
More informationHANKEL-NORM BASED INTERACTION MEASURE FOR INPUT-OUTPUT PAIRING
Copyright 2002 IFAC 15th Triennial World Congress, Barcelona, Spain HANKEL-NORM BASED INTERACTION MEASURE FOR INPUT-OUTPUT PAIRING Björn Wittenmark Department of Automatic Control Lund Institute of Technology
More informationA New Robust Decentralized Control Method for Interconnected Nonlinear Systems Based on State Extension and Adaptive Tracking
7 3rd International Conference on Computational Systems and Communications (ICCSC 7) A New Robust Decentralized Control Method for Interconnected Nonlinear Systems Based on State Etension and Adaptive
More informationRobust multi objective H2/H Control of nonlinear uncertain systems using multiple linear model and ANFIS
Robust multi objective H2/H Control of nonlinear uncertain systems using multiple linear model and ANFIS Vahid Azimi, Member, IEEE, Peyman Akhlaghi, and Mohammad Hossein Kazemi Abstract This paper considers
More informationRetrospective Cost Adaptive Control for Nonminimum-Phase Systems with Uncertain Nonminimum-Phase Zeros Using Convex Optimization
American Control Conference on O'Farrell Street, San Francisco, CA, USA June 9 - July, Retrospective Cost Adaptive Control for Nonminimum-Phase Systems with Uncertain Nonminimum-Phase Zeros Using Convex
More informationHigh Precision Control of Ball Screw Driven Stage Using Repetitive Control with Sharp Roll-off Learning Filter
High Precision Control of Ball Screw Driven Stage Using Repetitive Control with Sharp Roll-off Learning Filter Tadashi Takemura and Hiroshi Fujimoto The University of Tokyo --, Kashiwanoha, Kashiwa, Chiba,
More informationJ. Liang School of Automation & Information Engineering Xi an University of Technology, China
Progress In Electromagnetics Research C, Vol. 18, 245 255, 211 A NOVEL DIAGONAL LOADING METHOD FOR ROBUST ADAPTIVE BEAMFORMING W. Wang and R. Wu Tianjin Key Lab for Advanced Signal Processing Civil Aviation
More informationEnhancing Transient Response of Asymptotic Regulation with Disturbance Onset
211 American Control Conference on O'Farrell Street, San Francisco, CA, USA June 29 - July 1, 211 Enhancing Transient Response of Asymptotic Regulation with Disturbance Onset Kevin C. Chu and Tsu-Chin
More informationConditions for Suboptimal Filter Stability in SLAM
Conditions for Suboptimal Filter Stability in SLAM Teresa Vidal-Calleja, Juan Andrade-Cetto and Alberto Sanfeliu Institut de Robòtica i Informàtica Industrial, UPC-CSIC Llorens Artigas -, Barcelona, Spain
More informationTakagi Sugeno Fuzzy Sliding Mode Controller Design for a Class of Nonlinear System
Australian Journal of Basic and Applied Sciences, 7(7): 395-400, 2013 ISSN 1991-8178 Takagi Sugeno Fuzzy Sliding Mode Controller Design for a Class of Nonlinear System 1 Budiman Azzali Basir, 2 Mohammad
More informationAdaptive Control of a Class of Nonlinear Systems with Nonlinearly Parameterized Fuzzy Approximators
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 9, NO. 2, APRIL 2001 315 Adaptive Control of a Class of Nonlinear Systems with Nonlinearly Parameterized Fuzzy Approximators Hugang Han, Chun-Yi Su, Yury Stepanenko
More informationNeural Dynamic Optimization for Control Systems Part II: Theory
490 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL. 31, NO. 4, AUGUST 2001 Neural Dynamic Optimization for Control Systems Part II: Theory Chang-Yun Seong, Member, IEEE, and
More informationOPTIMAL CONTROL AND ESTIMATION
OPTIMAL CONTROL AND ESTIMATION Robert F. Stengel Department of Mechanical and Aerospace Engineering Princeton University, Princeton, New Jersey DOVER PUBLICATIONS, INC. New York CONTENTS 1. INTRODUCTION
More information/$ IEEE
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: EXPRESS BRIEFS, VOL. 55, NO. 9, SEPTEMBER 2008 937 Analytical Stability Condition of the Latency Insertion Method for Nonuniform GLC Circuits Subramanian N.
More informationCHATTERING-FREE SMC WITH UNIDIRECTIONAL AUXILIARY SURFACES FOR NONLINEAR SYSTEM WITH STATE CONSTRAINTS. Jian Fu, Qing-Xian Wu and Ze-Hui Mao
International Journal of Innovative Computing, Information and Control ICIC International c 2013 ISSN 1349-4198 Volume 9, Number 12, December 2013 pp. 4793 4809 CHATTERING-FREE SMC WITH UNIDIRECTIONAL
More informationThe ϵ-capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems
IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 11, Issue 3 Ver. IV (May - Jun. 2015), PP 52-62 www.iosrjournals.org The ϵ-capacity of a gain matrix and tolerable disturbances:
More informationDesign of Discrete-time Repetitive Control System Based on Two-dimensional Model
International Journal of Automation and Computing 9(2), April 212, 165-17 DOI: 1.17/s11633-12-629-1 Design of Discrete-time Repetitive Control System Based on Two-dimensional Model Song-Gui Yuan 1,2 Min
More informationIMECE NEW APPROACH OF TRACKING CONTROL FOR A CLASS OF NON-MINIMUM PHASE LINEAR SYSTEMS
Proceedings of IMECE 27 ASME International Mechanical Engineering Congress and Exposition November -5, 27, Seattle, Washington,USA, USA IMECE27-42237 NEW APPROACH OF TRACKING CONTROL FOR A CLASS OF NON-MINIMUM
More informationDecentralized Control of Nonlinear Multi-Agent Systems Using Single Network Adaptive Critics
Decentralized Control of Nonlinear Multi-Agent Systems Using Single Network Adaptive Critics Ali Heydari Mechanical & Aerospace Engineering Dept. Missouri University of Science and Technology Rolla, MO,
More informationNonlinear System Analysis
Nonlinear System Analysis Lyapunov Based Approach Lecture 4 Module 1 Dr. Laxmidhar Behera Department of Electrical Engineering, Indian Institute of Technology, Kanpur. January 4, 2003 Intelligent Control
More informationMANY algorithms have been proposed so far for head-positioning
26 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 10, NO. 1, FEBRUARY 2005 A Comparative Study of the Use of the Generalized Hold Function for HDDs Keitaro Ohno, Member, IEEE, Mitsuo Hirata, Member, IEEE,
More informationControlling Human Heart Rate Response During Treadmill Exercise
Controlling Human Heart Rate Response During Treadmill Exercise Frédéric Mazenc (INRIA-DISCO), Michael Malisoff (LSU), and Marcio de Queiroz (LSU) Special Session: Advances in Biomedical Mathematics 2011
More informationCharacterization of the stability boundary of nonlinear autonomous dynamical systems in the presence of a saddle-node equilibrium point of type 0
Anais do CNMAC v.2 ISSN 1984-82X Characterization of the stability boundary of nonlinear autonomous dynamical systems in the presence of a saddle-node equilibrium point of type Fabíolo M. Amaral Departamento
More informationH State-Feedback Controller Design for Discrete-Time Fuzzy Systems Using Fuzzy Weighting-Dependent Lyapunov Functions
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL 11, NO 2, APRIL 2003 271 H State-Feedback Controller Design for Discrete-Time Fuzzy Systems Using Fuzzy Weighting-Dependent Lyapunov Functions Doo Jin Choi and PooGyeon
More informationECE 516: System Control Engineering
ECE 516: System Control Engineering This course focuses on the analysis and design of systems control. This course will introduce time-domain systems dynamic control fundamentals and their design issues
More information