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Robustness

Robustness. Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo. Outline of Today’s Lecture. Review Important transfer functions Gang of Six Gang of Four Disturbance Rejection Noise Rejection Limitations Robustness

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Robustness

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  1. Robustness Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo

  2. Outline of Today’s Lecture • Review • Important transfer functions • Gang of Six • Gang of Four • Disturbance Rejection • Noise Rejection • Limitations • Robustness • Unmodeled dynamics • Tools: • Nyquist • Bode • Root locus • Course Review

  3. Sensitivity • Sensitivity is an evaluation of how the system responds to various signals compared to the design signal • In general, we want the system to respond to the reference input • We do not want the system to respond to noises and other signals that do not contribute to the accuracy of the desired output

  4. Several Transfer Functions N D Measurement Noise Disturbances u h Y U R E + + Y -1 Controller Process + + + +

  5. The Model N D u h R Y U E + + Y -1 Controller Process “Gang of Six” Complementary Sensitivity Function Load Sensitivity Function + + + “Gang of Four” + Noise Sensitivity Function Sensitivity Function

  6. Disturbance Rejection N D • We want our system designed such that the disturbances to the system are attenuated • Harold S. Black gave us the answer: negative feedback u h Y U R E + + Y -1 Controller Process + + + +

  7. Noise Rejection • We would also like noise rejection • Noise is most often high frequency signals caused by the sensors used to measure • Noise is presented as a result of the feedback terms • We do not have noise as defined here in an open system • In the closed loop error, noise is multiplied by T, the complementary sensitivity function, • In a system without a pre-filter, this is the transfer function • For this reason high frequency roll-off is important

  8. Limitations • Systems with right hand side poles and zeros are inherently hard to control • For a system with right hand side poles, pk, Bode showed that • Improvements in one frequency region are met with deteriorations in another frequency region • Sometimes called the waterbed effect

  9. Robustness • As we have said almost from the beginning, models are simplifications of the real object. • When we speak of robustness, we are speaking of the ability of our designed system to respond to flaws in our model • How well does the system respond if I did not model x correctly? • If I left something out, does the system still give an adequate response? (unmodeled dynamics) • If I modeled something incorrectly, does my system still respond as desired? (parameter uncertainty)

  10. Unmodeled Dynamics • In building a model we ignore a number of factors that • We think are not major factors in the performance • We do not know how to model effectively • We did not know about • These can act in three ways in our model P(s) P(s) P(s) Dm d D Additive Multiplicative Feedback

  11. Unmodeled Dynamics • One way to test how the unmodeled dynamics may effect the system is to assume that the plant transfer function iswhere P(s) is the simplified transfer function of the model and D are the unmodeled dynamics in terms of additive uncertainty. • We test robustness of the model using the tools that we have available to us (Nyquist plot, Bode plot and Root Locus) with assumed possible forms of D

  12. ExampleNyquist • Consider the pitch rate control on our aircraft with its controller: • If the unmodeled dynamics are stable (no rhs poles) thenwe have a circle in which the dynamics can act on the Nyquist diagram such that • Then T is a measure of relative robustness of the system The smaller the value of T the more robust the system C P + - sm |CD|

  13. ExampleBode We can show with the Bode the allowable uncertainty of the dynamics with regions

  14. ExampleRoot Locus • We can form a root locus for any parameter. • P(s)+D must produce a real response • When drawing the root locus, we solve the characteristic equation: • Therefore, we need to separate D from the remainder of the form to plot: Del Positive Unstable Region Del Negative

  15. Course Summary • Modeling • State Space Formulation • Stability • Modes • Reachability/Constructability • State Feedback Compensation • Observability • Transfer Functions • Block Diagrams • Root Locus • Nyquist Stability Analysis • Bode Plots • PID Control • Loop Shaping • Sensitivity • Robustness

  16. Where Do You Find Controls? • Everywhere!

  17. Open Loop Control • Usually “set point” systems • Advantages • Simple • Sensitive to environment • Set and forget • Disadvantages • Non correcting • Sensitive to disturbances • Insensitive to environment • Examples • Irrigation systems • Washing machines Sensing Compute Actuate

  18. Closed Loop Control Actuate Sense • Adds a feedback loop to the control system • For computational purposes, it is shown as Controller Plant Compute Sensor Disturbance + or - + or - Output Input + or - + or -

  19. Basic Control Actions • Bang-Bang (Off-On) • Fixed two state or multistate control actions • Control question: how to chose? • Proportional • Control in proportion to error • Integral • Control based on size and duration of error • Derivative • Control based on size and change of error • Combined (PID) • All three: Proportional, Integral and Derivative • Most used

  20. Models REAL WORLD OBSERVATIONS SENSE FORMULATE TEST EXPLANATION/ PREDICTION MATHEMATICAL MODEL INTERPRET

  21. Engineering Modeling Procedure • Understand the problem • What are the factors and relevant relationships? • What assumptions can be made? • What equilibrium conditions exist? • What should the result look like? • Draw and label an engineering sketch • Free body diagram • Hydraulic schematic • Electrical schematic • Write the equilibrium equations (usually differential or difference) • Newton 2nd Law • Kirchoff Laws for current and voltages • Flow continuity laws • Solve the equations for the desired result • Check the validity of the results

  22. Distributed vs. Lumped Parameters • Distributed parameter • Analysis is at the material element level • Partial differential equations describe the transfer of force from the constitutive equations • FEM/BEM often used • Lumped parameter • Analysis is at the component level • Component properties are self contained and complete • ODE/Diff E based on linking component parameters • Equations solved analytically or numerically

  23. State Space FormulationContinuous Models • Let x be a vector formed of the state variables • The number of components of the state vector is called the order • Formulate the system as • The matrices A, B, C and D have constant elements • The matrix A is the called the State Dynamics Matrix • The matrix B is called the Input or Control Matrix • The matrix C is called the Output or Sensor Matrix • The matrix D is called the Pass Through or Direct term

  24. State Space FormulationDiscrete Models • Let x be a vector formed of the state variables • The number of components of the state vector is called the order • Formulate the system as • The matrices A, B, C and D have constant elements • The matrix A is the called the State Dynamics Matrix • The matrix B is called the Input or Control Matrix • The matrix C is called the Output or Sensor Matrix • The matrix D is called the Pass Through or Direct term

  25. State Space Formulation • Procedure: • Develop the equations of equilibrium • Put the equilibrium equations in the form of the highest derivative equal the remainder of the terms • Make a choice of states, the input and the outputs • Write the equilibrium equations in terms of the state variables • Construct the dynamics, the input, the output and the pass through matrices • Write the state space formulation

  26. Simulink

  27. Two Mathematical Problems Frequently Encountered in Controls • Find the roots of an equation • Methods • Trial and Error (bracketing methods add a bit of science to this) • Graphics • Closed form solutions (e.g.: quadratic formula) • Newton Raphson • Find the solution at a given time for given conditions • Various differential and difference equations analytic solutions (sometimes reformulated as find the roots problem) • Numerical Methods • Newton Cotes Methods (trapezoidal rule, Simpson’s rule. etc. for integration) • Euler’s Method • RungaKutta/Butcher Methods • Many other techniques (Adams-Bashforth, Adams-Milne, Hermite–Obreschkoff, Fehlberg, Conjugate Gradient Methods, etc.)

  28. Numerical Methods • Numerical methods follow the procedure • Step1: Initialize: Select some initial value • Step2: Estimate using (guess, some analytical technique) a new value at increment “i” • Step 3: Is the system converging? If not, use something else. We usually know a priori whether a method will converge or not form mathematics. Therefore, this step is often omitted. • Step 4: Is the change from the previous value to current value smaller than our acceptable error? • If not, make the current value the previous value and return to step 2. • If so, stop and accept the new value as the solution.

  29. Newton Raphson Method for finding roots • Probably the most common numerical technique • simple • efficient • flexible • It can be shown from a truncated Taylor’s Series that • Provided that the slope at the test points is consistent, we can iterate to a solution within our error tolerance f(t) f(ti) ti+1 ti Problems occur if the slope reverses sign such as in an oscillation or becomes very flat t

  30. RungeKutta/Butcher Method • Has its origins in a 2 variable Taylor Series Expansion • The function is called the increment function • RK4 is a four factor expansion of the incrementing function • For RK4: • Butcher’s method uses 5 factors is more accurate than RK4 at a given time step

  31. 2nd Order System Response z z z

  32. System Response: Step Input • The time history of a system’s outputs Often called the system path, trajectory or time series { Overshoot Mp Steady State Rise time, tr Transient period=settling time, ts

  33. System Response: Frequency Response • Time history with respect to a sinusoid: Phase Shift, DT Amplitude Ay Amplitude Au Input Sin(t) Period,T Transient Response

  34. Determination of Stabilityfrom Eigenvalues

  35. Modes • Each eigenvalue is associated with a mode of a system • Each eigenvalue is associated with an eigenvector, , such that • If the eigenvalues are distinct, we can form the modal matrix, M, from the eigenvectors and use it to diagonalize the dynamics matrix A which will then separate each mode in the form of a differential equation: • When a set of eignevectors are repeated (equal to each other) a full set of n linear independent eignevectors may or may not exist. In that case we need to form the Jordan blocks for the repeated elements

  36. Transformations • Say we have some matrix T that is invertible (this is important) which results in the vector z when x is premultiplied by T. We then say that we have transformed the vector x into z, or alternatively, we have transformed x into z:

  37. Convolution Equation is called the “Convolution Equation” Expresses the effect of an input on the system • What is convolution? • a twisting or folding together of two things • A convolution is found in many phenomena: • A sound that bounces off of a wall and interacts with the source sound is a convolution • A shadow is a convolution between the light source and the object producing the shadow • In statistics, a moving average is a convolution

  38. System Response • Another common test function is a sinusoid for frequency response • Since we have a linear system, we only need and assuming that the eigenvalues A do not equal s } } Steady State Transient

  39. Linearization Techniques • Ignore the nonlinearity • In some cases, the nonlinearity has a relatively small effect • In those cases, build a linear system and treat the nonlinearity as a disturbance • Small angle approximations • Often only useful near equilibrium points • Taylor Series Truncation about an operating point • Assumes that 2nd and higher orders are negligible • Feedback linearization 0

  40. Reachability • We define reachability (often times called controllability) by the following: • A state in a system is reachable if for any valid states of the system, say, initial state at time t=0, x0 , and a state xf, there exists a solution for t>0 such that x(0) = x0 and x(t)=xf. • There are systems which we can not control • the states are not reachable with our input. • There in designing control systems, it is important to know if the system is controllable. • This is closely linked with the concept of ergodicity of the system in which we ask the question whether or not it is possible to with some measure of our system to measure every possible state of the system.

  41. Reachability • For the system, , all of the states of the system are reachable if and only if Wr is invertible where Wr is given by

  42. Reachable Canonical Form • A system is in the reachable canonical form if it has the structure Such a structure can be represented by blocks as … y S S S S D c1 c2 cn-1 cn … z1 z2 zn zn-1 u S -1 a1 a2 an-1 an … S S S

  43. Control System Objective Given a system with the dynamics and the output Design a linear controller with a single input which is stable at an equilibrium point that we define as

  44. Our Design Structure Disturbance Controller u Plant/Process Input r Output y S S kr State Controller Prefilter x -K State Feedback

  45. 2nd Order Response • As the example showed, the characteristic equation for which the roots are the eigenvalues allow us to design the reachable system dynamics • When we determined the natural frequency and the damping ration by the equationwe actually changed the system modes by changing the eigenvalues of the system through state feedback wn=1 z=0.6 Im(l) Im(l) x wn=4 1 1 z=0.1 x x x wn=2 z=0.4 x z=0 x x wn=1 z=0.6 Re(l) Re(l) z=1 z=1 x x wn -1 -1 z z=0.6 wn=1 x x z=0 x wn=2 x z=0.4 x x -1 -1 z=0.1 x wn=4

  46. State Feedback Design with the Reachable Canonical Equation • Since the reachable canonical form has the coefficients of the characteristic polynomial explicitly stated, it may be used for design purposes:

  47. Observability • Can we determine what are the states that produced a certain output? • Perhaps • Consider the linear system We say the system is observable if for any time T>0 it is possible to determine the state vector, x(T), through the measurements of the output, y(t), as the result of input, u(t), over the period between t=0 and t=T.

  48. Observers / Estimators Input u(t) Output y(t) Noise State Observer/Estimator

  49. Testing for Observability • For x(0) to be uniquely determined, the material in the parens must exist requiring to have full rank, thus also being invertible, the common test • Wo is called the Observability Matrix

  50. Observable Canonical Form • A system is in Observable Canonical Form if it can be put into the form Where ai are the coefficients of the characteristic equation … u bn bn-1 b2 b1 D y z2 zn zn-1 … z1 S S S S S an an-1 a2 a1 -1 …

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