# Lecture 12. Upcoming labs: Final Exam on 12/21/2015 (Monday)10:30-12:30

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1 289 Upcoming labs: Lecture 12 Lab 20: Internal model control (finish up) Lab 22: Force or Torque control experiments [Integrative] (2-3 sessions) Final Exam on 12/21/2015 (Monday)10:30-12:30 Today: Recap of Internal Model Control Systems and Control Review Servo valve modeling TA evaluation (5 mins)

2 Electrohydraulic Force/Torque Control 290 Objective: Accurately apply predefined force/torque (stress) trajectories to specimen Often until fails

3 291 Setup and Procedures Linear system: Actuator pushing against a leaf spring (one end constraint). Force measurement by load cell. Rotary system: Actuator torquing an aluminum rod. Torque measurement by torque cell. It is a new system! Expect some nonlinearity of the spring Apply all your knowledge!

4 292 Objectives: Design and implement controllers to accurately track different types of trajectories Steps: 1. System identification (valve command input, force/torque output) 2. Choose appropriate controllers for the trajectories (steps, biased sinusoids, triangular wave) 3. Analyze and design controllers 4. Implement control 5. Go to steps 2/3 and improve performance

5 293 Internal Model Control Proportional-Integral control excellent for canceling constant disturbance or tracking constant command Generalize idea for other disturbances and commands, such as Sine/cosines, ramps, or other polynomials? Recall for P-I: C(s) = Kp + KI/s P-I control can generate constant input (u) even as error e(t) à 0 For other types of disturbances, the internal model control should generate input to cancel out disturbances.

6 Internal Model Control - Architecture Sine/cosine disturbance: U(s) = C(s) E(s) Suppose error converges to 0 so using partial fraction, So u(t) will generate some sinusoid/cosine term with frequency omega Cf. with integral control.. s on denominator in P-I that generates constants

7 295 Internal Model Control - Architecture 2. Polynomials disturbances? 1, t, t 2, etc.? 3. Combinations of sinusoids and polynomials? 4. Trajectory tracking instead of disturbance rejection? To track sinusoids need sinusoidal inputs To track polynomials, need polynomials as inputs (check constant case)

8 296 Assigning Closed Loop Poles The above suggests the form (denominator) of C(s) for various disturbances How to pick numerator of C(s)? Choose closed loop poles and use numerator to achieve target pole locations What are desirable closed loop pole locations? E.g. G(s) = 2/(s+3); D(s) = sinusoids + constant Etc

9 297 Comparison between controllers P simple, need large Kp for good performance P-I regulate constant command (or ramp for integrator plants) and rejecting constant disturbance; Values of disturbance or command no needed IMC track or reject sinusoids or polynomials Values of disturbance or command no needed Need only the type Feedforward Arbitrary command trajectories Can combine with feedback control, e.g. P, P-I, or IMC

10 Feedforward Example 298 Supposed a closed loop system has been designed, we think it has a transfer function: Ĝ c (s) = 25 s + 25 Design a feedforward controller such that the output y(t) track an arbitrary trajectory r(t). Write it out in as sum of differentiators and proper transfer function. If the actual closed loop transfer function is: G c (s) = 20 s + 20 How would it change its ability to track sinusoids for different frequencies?

11 If a plant is a first order system IMC Example G(s) = 2 s Write down the form of the Internal Model Controllers if: r(t) =a + bt+ ccos(3t + d)+e sin(7t) d(t) =g + hcos(2t) How to find the coefficients of the IMC controller?

12 300 Objectives Introduce fluid power component, circuits, and systems Functions, modeling and analysis Provide hands on experience in designing, analyzing and implementing control systems for real and physical systems; Consolidate concepts in Systems Dynamics/Control (ME3281) modeling, control and other dynamical systems Course syllabus, lab assignments, notes, etc. on course webpage (subject to change without notice)

13 301 Expected Outcome Familiarity with common hydraulic components, their use, symbols, and mathematical models Ability to formulate / analyze math models for simple hydraulic circuits Comfortable with commercial hydraulic catalogs Ability to identify single input single output (SISO) dynamical systems Ability to design, analyze and implement simple control systems Appreciation of advantages and disadvantages of various types of controllers Ability to relate control systems analysis with actual performance Intuitive and mathematical appreciation of dynamical system concepts (e.g. stability, instability, resonance) Appreciation of un-modeled real world effects Become very familiar with using Matlab for analysis and plotting.

14 Critical Basic Concepts 302 Transfer function Input-output relationship Block diagram à transfer function Closed loop pole locations and characteristics of response Stability Steady state response via final value theorem Frequency response

15 Critical Controls Concepts 303 Control system objectives: Stability: Determined by closed loop pole location (Reference Tracking) Performance: Robustness to disturbance Insensitivity to model uncertainty Immunity to measurement noise

16 Feedback versus feedforward 304 Feedback control Advantages: Compensates for disturbances and model uncertainty Disadvantages: Can be unstable if not designed correctly Usually cannot track ARBITRARY reference trajectories PEFECTLY Feedforward control Advantages: Perfect tracking for ARBITRARY reference trajectories! Disadvantages: Cannot compensate for disturbances or model uncertainty Feedback and feedforward control can be combined!!!! TRY it for your lab 22! Feedforward keeps error small so higher feedback gains are possible

17 Comparison of Feedback Controllers Proportional Control 305 Advantage: Simple Disadvantages: Need infinity gain to good performance, Increases gain in all frequencies Compromise with noise and robustness, Steady error with constant disturbances or ramp (and step in general) inputs

18 306 Proportional-Integral Control Advantages: Zero-steady state error for step (and ramps in general) references and disturbances Increases low frequency gain while keeping high frequency gain low Steady state error relatively insensitive to model uncertainty Disadvantages: Works only for limited set of reference trajectories and disturbances 2 gains to tune 2 nd order closed loop system (with 1 st order plant) à possibility of resonance, under-damped etc. Good for situations when required control input (in steady state) is a constant

19 307 Advantage: Internal Model Control (Generalized P-I) Zero-steady state error for step, ramps, sinusoids, exponential etc. references and disturbances Increases gain at the specific frequency of references while keeping gains at other frequencies low Insensitive to model uncertainty as long as closed loop is stable Disadvantage: Works only for limited types of reference trajectories and disturbances Many gains to tune Complex needs to rely on analysis

20 Control Design Procedures 1. What is the system being controlled? Model it System identification Choose the type of controller P, PI, IMC, Feedforward etc. 3. Formulate closed loop transfer function, and analyze performance 4. Design desired pole locations (where should they be?) 5. Calculate the controller gains to obtain the poles 6. Add feedforward control

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