Robustness Professor Walter W. Olson

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

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

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

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

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

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

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

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

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)

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) D P(s) d P(s) Dm Additive Multiplicative Feedback

Unmodeled Dynamics One way to test how the unmodeled dynamics may effect the system is to assume that the plant transfer function is where 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

Example Nyquist Consider the pitch rate control on our aircraft with its controller: If the unmodeled dynamics are stable (no rhs poles) then we 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|

Example Bode We can show with the Bode the
allowable uncertainty of the dynamics with regions

Example Root 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

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

Where Do You Find Controls?
Everywhere!

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

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

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

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

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

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

State Space Formulation Continuous 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

State Space Formulation Discrete 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

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

Simulink

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 Runga Kutta/Butcher Methods Many other techniques (Adams-Bashforth, Adams-Milne, Hermite–Obreschkoff, Fehlberg, Conjugate Gradient Methods, etc.)

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.

Newton Raphson Method for finding roots
f(t) f(ti) ti ti+1 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 Problems occur if the slope reverses sign such as in an oscillation or becomes very flat

Runge Kutta/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

2nd Order System Response
z

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

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

Determination of Stability from Eigenvalues
Continuous Time Discrete Time Unstable Stable Asymptotic Stability

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

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:

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

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 } } Transient Steady State

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

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.

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

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 D c1 c2 cn-1 cn -1 a1 a2 an-1 an S … u y z1 z2 zn-1 zn

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

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

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 equation we actually changed the system modes by changing the eigenvalues of the system through state feedback z -1 1 Re(l) Im(l) x z=1 z=0.6 z=0.4 z=0.1 z=0 wn=1 wn -1 1 Re(l) Im(l) x z=0.6 wn=1 wn=2 wn=4

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:

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.

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

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

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 … S bn bn-1 b2 b1 D an an-1 a2 a1 -1 u z2 zn-1 z1 zn y

Dual Canonical Forms

Observers / Estimators
Observer/Estimator Input u(t) Output y(t) Noise State B C A L + _ u y

Alternative Method of Analysis
Up to this point in the course, we have been concerned about the structure of the system and described that structure with a state space formulation Now we are going to analyze the system by an alternative method that focuses on the inputs, the outputs and the linkages between system components. The starting point are the system differential equations or difference equations. However this method will characterize the process of a system block by its gain, G(s), and the ratio of the block output to its input. Formally, the transfer function is defined as the ratio of the Laplace transforms of the Input to the Output:

System Response From Lecture 11
We derived for } } Transient Steady State Transfer function is defined as

Linear System Transfer Functions
General form of linear time invariant (LTI) system is expressed: For an input of u(t)=est such that the output is y(t)=y(0)est Note that the transfer function for a simple ODE can be written as the ratio of the coefficients between the left and right sides multiplied by powers of s The order of the system is the highest exponent of s in the denominator.

Block Algebra G x Gx G x Gx G x H Hx + - Gx (G-H)x G-H (G-H)x x G Gx-z
Gz G(x-z) G G(x-z) + - x z

Loop Analysis (Very important slide!)
H(s) + - R(s) Y(s) E(s) B(s) Negative Feedback G(s) Positive Feedback H(s) + R(s) Y(s) E(s) B(s)

Gain, Poles and Zeros The roots of the polynomial in the denominator, a(s), are called the “poles” of the system The poles are associated with the modes of the system and these are the eigenvalues of the dynamics matrix in a state space representation The roots of the polynomial in the numerator, b(s) are called the “zeros” of the system The zeros counteract the effect of a pole at a location The variable s is a complex number: The value of G(0) is the zero frequency or steady state gain of the system

Root Locus The root locus plot for a system is based on solving the system characteristic equation The transfer function of a linear, time invariant, system can be factored as a fraction of two polynomials When the system is placed in a negative feedback loop the transfer function of the closed loop system is of the form The characteristic equation is The root locus is a plot of this solution for positive real values of K Because the solutions are the system modes, this is a powerful design tool While we focus here on the gain, K, we can plot any parameter this way

Plotting a Transfer Function Root Locus
The path is determined from the open loop transfer function by varying the gain ‘s’ as used in a transfer function is a complex number Poles will be marked with X Zeros with be marked with an O Each path represents a branch of the transfer function in the complex plane All paths start at poles and end at zeros There must be a zero for each pole Those that are not shown on the plot are at infinity Matlab command rlocus(sys)

Reading the Bode Plot Amplifies Attenuates Input Response
Note: The scale for w is logarithmic The magnitude is in decibels Amplifies Attenuates decade also, cycle Input Response

What is a decibel? The decibel (dB) is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities. (IEEE Standard 100 Dictionary of IEEE Standards Terms, Seventh Edition, The Institute of Electrical and Electronics Engineering, New York, 2000; ISBN ; page 288) Because decibels is traditionally used measure of power, the decibel value of a magnitude, M, is expressed as 20 Log10(M) 20 Log10(1)=0 … implies there is neither amplification or attenuation 20 Log10(100)= 40 decibels … implies that the signal has been amplified 100 times from its original value 20 Log10(0.01)= -40 decibels … implies that the signal has been attenuated to 1/100 of its original value

Frequency Response General form of linear time invariant (LTI) system was previously expressed as We now want to examine the case where the input is sinusoidal. The response of the system is termed its frequency response.

Phase As with magnitude there are 4 factors to consider which can be added together for the total phase angle. We will consider, in turn, The sign will be positive if the factor is in the numerator and negative if the factor is in the denominator

Matlab Command bode(sys)

Laplace Transform Traditionally, Feedback Control Theory was initiated by using the Laplace Transform of the differential equations to develop the Transfer Function The was one caveat: the initial conditions were assumed to be zero. For most systems a simple coordinate change could effect this If not, then a more complicated form using the derivative property of Laplace transforms had to be used which could lead to intractable forms While we derived the transfer function, G(s), using the convolution equation and the state space relationships, the transfer function so derived is a Laplace Transform under zero initial conditions

Laplace Transform CAUTION: Some Mathematics is necessary!
The Laplace transform is defined as Fortunately, we rarely have to use these integrals as there are other methods

Properties of the Laplace Transform
Laplace Transforms have several very import properties which are useful in Controls Now, you should see the advantage of having zero initial conditions

Final Value Theorem If f(t) and its derivative satisfy the conditions for Laplace Transforms, then This theorem is very useful in determining the steady state gain of a stable system transfer function Do not apply this to an unstable system as the wrong conclusions will be reached!

Loop Nomenclature Disturbance/Noise Reference Error Input signal
+ - Output y(s) Error signal E(s) Open Loop Signal B(s) Plant G(s) Sensor H(s) Prefilter F(s) Controller C(s) Disturbance/Noise Reference Input R(s) The plant is that which is to be controlled with transfer function G(s) The prefilter and the controller define the control laws of the system. The open loop signal is the signal that results from the actions of the prefilter, the controller, the plant and the sensor and has the transfer function F(s)C(s)G(s)H(s) The closed loop signal is the output of the system and has the transfer function

Open Loop System Nyquist Plot
Error signal E(s) + Output y(s) Open Loop Signal B(s) Plant P(s) Controller C(s) Input r(s) Sensor -1 Imaginary B(-iw) Plane of the Open Loop Transfer Function -1 B(0) Real B(iw) -1 is called the critical point

Simple Nyquist Theorem
Error signal E(s) + Output y(s) Open Loop Signal B(s) Plant P(s) Controller C(s) Input r(s) Sensor -1 -1 Real Imaginary Plane of the Open Loop Transfer Function B(0) B(iw) -1 is called the critical point Stable Unstable -B(iw) Simple Nyquist Theorem: For the loop transfer function, B(iw), if B(iw) has no poles in the right hand side, expect for simple poles on the imaginary axis, then the system is stable if there are no encirclements of the critical point -1.

Full Nyquist Theorem Assume that the transfer function B(iw) with P poles has been plotted as a Nyquist plot. Let N be the number of clockwise encirclements of -1 by B(iw) minus the counterclockwise encirclements of -1 by B(iw)Then the closed loop system has Z=N+P poles in the right half plane. Show with Sisotool

Margins Margins are the range from the current system design to the edge of instability. We will determine Gain Margin How much can gain be increased? Formally: the smallest multiple amount the gain can be increased before the closed loop response is unstable. Phase Margin How much further can the phase be shifted? Formally: the smallest amount the phase can be increased before the closed loop response is unstable. Stability Margin How far is the the system from the critical point?

Gain and Phase Margin Definition Nyquist Plot
-1

Gain and Phase Margin Definition Bode Plots
Magnitude, dB Positive Gain Margin w Phase, deg -180 Phase Margin w Phase Crossover Frequency

Non-Minimum Phase Systems
Non minimum phase systems are those systems which have poles on the right hand side of the plane: they have positive real parts. This terminology comes from a phase shift with sinusoidal inputs Consider the transfer functions The magnitude plots of a Bode diagram are exactly the same but the phase has a major difference:

The Ideal PID Controller
The input/output realtionship for the PID Controller is the Integral-Differential Equation The ideal PID controller has the transfer function Structurally it would look like + -1

Ziegler-Nichols PID Tuning Method 1 for First Order Systems
The advice given is to draw a line tangent to the response curve through the inflection point of the curve. However, a mathematical first order response doesn’t have a point of inflection as it is of the form (at no place does the 2nd derivative change sign.) My advice: place the line tangent to the initial curve slope You also have to adjust for the gain K of the system by multiplying compensator by 1/K Lag L Rise Time T Type kp Ti Td P PI PID

Error signal E(s) + Output y(s) Open Loop Signal B(s) Plant P(s) Controller C(s) Input r(s) Sensor -1 Loop Shaping We have seen that the open loop transfer function, has profound influences on the closed loop response The key concept in loop shaping designs is that there is some ideal open loop transfer (B(s)) that will provide the design specifications that we require of our closed loop system Loop shaping is a trial and error process: Everything is connected and nothing is independent What we gain in one area may (usually?) causes loss in other areas Often times, out best controller is a compromise between demands To perform loop shaping we can used either the root locus plots or the Bode plots depending on the type of response that we wish to achieve We have already considered an important form of loop shaping as the PID controller

Lead and Lag Compensators
The compensator with a transfer function is called a lead compensator if a<b and a lag compensator if b>a The lead and the lag compensator can be used together Note: the compensator does add a steady state gain of that needs to be accounted for in the final design There are analytical methods for designing these compensators (See Ogata or Franklin and Powell)

Example We achieved the specifications once the pole of the compensator was moved out to -9 and we adjusted the gain for the 0.6 damping.

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

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

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

Summary Robustness Unmodeled dynamics Tools: Course Review Nyquist
Bode Root locus Course Review Next: Final Exam

Robustness Professor Walter W. Olson

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