Copyright. SRS, U DuE, rof. Söffker. Course Control Theory WiSe 2014/15

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1 Course Theory WiSe 2014/15 Room: SG 135 Time: Fr pm (lecture and exercise) Practical exercise: 2nd part of semester Assistants: Xi Nowak, M.Sc.; WEB: Manuscript Note 1: Note 2: The collected material is prepared for the use only in connection with the lecture. It is not allowed to use this material outside the lecture of Prof. Söffker. The reprinted figures are coming if nothing different is mentioned - from the textbook of Prof. Lunze and are free for use in connection with this textbook-based course. Course theory LU-0: Preliminary remarks rof. Söffker 1/10

2 Content of the course - Requirements -Contents - Relations to other lectures of the chair (Bachelor, Master) Idea behind the the course - Manusscript presented with tablet pc (#former semesters) - Text book >>Ogata/Lunze (Library) Theoretical and practical exercises - theoretical exercise: included within the course - practical exercise: 3 practical each of 4 hours in June - add. Exercise about control technique contents May 2nd-6th Consulting hours - Th am Exercises tasks - Lunze / Ogata /others Exam >> as usual (2 hours written exam in february/march 2015) Course theory LU-0: Preliminary remarks rof. Söffker 2/10

3 C o SR py r S i ro, gh f. U t Sö D u ffk E, er 0.2 Add. remarks I 3/10 Course theory LU-0: Preliminary remarks

4 0.2 Add. remarks Ib Course theory LU-0: Preliminary remarks rof. Söffker 4/10

5 C o SR py r S i ro, gh f. U t Sö D u ffk E, er 0.2 Add. remarks Ic 5/10 Course theory LU-0: Preliminary remarks

6 C o SR py r S i ro, gh f. U t Sö D u ffk E, er 0.2 Add. remarks Id 6/10 Course theory LU-0: Preliminary remarks

7 0.2 Add. remarks II Course theory LU-0: Preliminary remarks rof. Söffker 7/10

8 0.2 Add. remarks IIb Course theory LU-0: Preliminary remarks rof. Söffker 8/10

9 0.2 Add. remarks IIc Course theory LU-0: Preliminary remarks rof. Söffker 9/10

10 Course theory LU-0: Preliminary remarks rof. Söffker 10/10

11 LU-1 Introduction to MIMO systems What is the content of lecture units 1 and 2? >> You should learn about the prerequisites of the course, especially about some terms and methods to use the time to exam to learn from books if neccessary. >> You should get sensible for the MIMO topic. > This is illustrated with static and dynamic examples. >> The differences between SISI and MIMO are declared. > Representation, Methods >> The main differences between SISO and MIMO are developed by changing the viewpoint from SISO to MIMO > State space representation, Frequency domain, Couplings of elements, Time behavior and integration of. Söffker Course theory LU-1-2: Introduction, description, behavior I

12 LU-1 Introduction to MIMO systems 1.1 Characterisctics of MIMO systems (MIMO: Multi Input - Multi Output) Inputs and Outputs: Complex dynamics: Coupled systems: Result: Practical examples I: - Heat exchanger of. Söffker Course theory LU-1-2: Introduction, description, behavior 1/9

13 LU-1 Introduction to MIMO systems Practical examples II: - Elastic beam (robot beam) of. Söffker Course theory LU-1-2: Introduction, description, behavior 2/9

14 1.2 Representation of values - m-dim. input vector u(t) - r1-dim. output vector y(t) - r2-dim. vector of reference w(t) - p-dim. disturbance vector d(t) > written form > spoken form > A special case: the SISO system of. Söffker Course theory LU-1-2: Introduction, description, behavior 3/9

15 1.3 Analysis and control goals in the context of MIMO systems - SISO control design goals : stability, reference control, disturbance compensation, dynamic requirements plus - Analysis of the dynamics Goal: Use of couplings > decoupling - Analysis of the inner couplings What acts how? (- Guaranty of the functionality of the control loop) Robustness against failures of actuators and sensors) 1.4 Transformation/Use of the SISO methods to analysis and synthesis of MIMO-systems possible? Description (state space, transfer function, weighting function ) - Description tools to describe the dynamic behavior (poles, zeros, Hurwitz ) - Signal models (internal model principle ) ( works for single transfer pathes) - Root locus - Frequency domain-based control approaches ( but: more complex relations) 1.5 New ideas and methods: - Analysis of internal couplings - Design of control structures using internal couplings Course theory LU-1-2: Introduction, description, behavior 4/9 of. Söffker

16 LU-2 Description and behavior of linear MIMO systems 2.1 Description techniques for SISO systems - Differential equations - Transfer functions - State space description 2.2 Description techniques for MIMO systems - Set of linear differential equations > Vector differential equation of. Söffker Course theory LU-1-2: Introduction, description, behavior 5/9

17 - Canonical regular form (diagonal canonical form) > (effect of inputs, effect of outputs, modal measurements) > Calculating modes (eigenvalue equation: eigenvalues, eigenvector equation: eigenvectors) > Decomposing the system (calculating the modes) of. Söffker Course theory LU-1-2: Introduction, description, behavior 6/9

18 2.3 Description of the MIMO-system in frequency domain - Transfer function matrix (be careful > MIMO-case) - Frequency domain description of a MIMO systems > elementwise analog to the SISO case of. Söffker Course theory LU-1-2: Introduction, description, behavior 7/9

19 - Connections between the frequency domain and the state space > Transfer behavior in state space: Rosenbrock system matrix of. Söffker Course theory LU-1-2: Introduction, description, behavior 8/9

20 2.4 Parallel and series network of MIMO transfer elements 2.5 (Time-)behavior of MIMO systems (of the state-space description) - Equations of motion - Output equations of motion - Instationary (transient) and stationary (# constant) behavior of. Söffker Course theory LU-1-2: Introduction, description, behavior 9/9

21 What you should learm from LU 1-2: >> The fact that MIMO systems link several inputs and outputs with one another leads to complex dynamics, i.e. based on the dynamic couplings. From this it results that the known fast controller design strategy (SISO: PID-approaches etc.), which are used in SISO approaches and which realizes a simple connection between the controller input and system output, can not be used with MIMO systems. >> A system analysis (stability, modal characteristics, observability, controllability) always precedes the controller design (synthesis) in the case of MIMO systems. >> The state space representation permits a systematic representation of all linear MIMO systems. The analysis methods are independent from the number of the states as well as from the dynamics. >> The transfer function matrix shows the dynamic effects between the several inputs and outputs. >> The structural properties are shown in the time domain using the state space representation, in the frequency domain using the Rosenbrock matrix. >> The state space representation is the linear and explicit special case of the general representation for the time integration of dynamic systems (in explicit form). of. Söffker Course theory LU-1-2: Introduction, description, behavior S

22 University of Duisburg-Essen Univ.-Prof. Dr.-Ing. Course Theory/ Technique Spring/Winter 2014/15 Terms and Definitions System: Purpose-oriented part of the real world. The system is in interaction with the real world, whereby from the real world inputs are acting to the system and outputs are acting from the system to the environment. Physical variables / physical values: Interaction qualities (qualities / states) within a system or from the system to the environment, in technical systems usually of physical nature. Physical values are usually defined in connection with the purpose of related systems to be considered. The variables to be considered in control are usually scalars or vectors. The main property of variables considered within system dynamics or control is the time depending behavior. variable / Output (single input single output system): Variable of the system to be controlled. The variable should be usually fix or variable. In Single Input - Single Output (SISO) systems the output (here only one exists) is the control variable. Open loop system: Open loop systems are working so that the inputs of the systems are affecting the outputs depending on the dynamic properties of the system and not vice versa. The characteristic property of an open loop system is the non closed behavior of the signal flow. Closed loop system: Within a closed loop system the control variable is usually online measured and compared with a given desired / reference variable (which also can be zero). The result is given to the input in the way that it is acting so that the control variable tends to the desired value. This realizes the feedback and closes the loop. The characteristic property of a closed loop system is the closed behavior of the signal flow. This changes the dynamic behavior of the overall system and allows the automatic compensation of disturbances acting to the system and affecting the control variable. Disturbance (variable / value): From the environment to the system acting variable, which influences the control variable in an unwanted way. Input (variable / value): Variable acting direct to the system and changes the output of the system. In open loop system the input variable acts from the environment to the system, in closed loop systems the input is the output of the controller and the input of the system to be controlled. Desired variable /value / reference variable/value: Variable from the environment to the controller, which gives the reference for the output as the value to be controlled. deviation / control difference: Variable denoting the difference between the desired and the control variable. The control deviation is an internal variable inside the controller.

23 Plant (system to be controlled): The plant is the system to be controlled (closed loop) or the system to be affected by the input (open loop). ler: The controller is the part of the system, which affects the plant using an actuator. From a practical point of view controller and actuator are distinguished, from a theoretical point the control unit combines the control and the actuator and is called controller. From a practical point of view the technical control unit often also consists of the unit for comparison of desired and control variable. Transfer element: Abstract understanding of a general system with dynamic properties, which can be the system to be controlled or the controller or another element of the transfer chain. The important aspect is that the transfer element is a system, whereby the dynamic properties of the transmittance from input to output are of interest. The relevant aspect is the dynamic characterization of the transfer behavior. Plant and controllers are in this way nothing more than transfer elements, the denotation plant or controller results from the function of the transfer element within the loop. Transfer system: Abstract understanding of a set of elements building up a new element > system. Actuator: Actuator is the physical device in front of the plant used for physical excitation of the plant to affect the plant. In the theoretical / abstract consideration the actuator is part of the transfer element controller. Measurement device: The physical device used for measuring of the output / the control variable is called measurement device / unit. In the theoretical / abstract consideration the dynamical properties of usual neglected, so the measurement device is understand is an ideal transfer element without any kind of dynamic properties. For practical considerations the dynamics properties of the measurement unit as well as of the actuator has to be considered. Reference value: Parameter of the reference variable value: Parameter of the control variable Remark: The given terms and definitions are used from different authors etc. in a different way. The important aspect is the verbal/term-based separation from signals and physical variables, from systems and behaviors, from the physical/technical realization and from the mathematical abstract mapping. For example equations are used to described the behavior of systems, the output variables of technical systems show also this behavior. In older text books and practical oriented literature this distinction is often not realized. It is a good idea to understand the clear separation between reality, the mapping-based mathematical representation to be open for future advanced information science oriented control and interaction approaches.