1. Module 2: Representing Process and Disturbance Dynamics Using Discrete Time Transfer Functions

2. chee825 - Winter 2004J. McLellan2 Dynamic Models - a First Pass establish linkage between process dynamic representations and possible disturbance representations key concept - dynamic element, represented by a transfer function, driven by random shock sequence »IID Normal - white noise

3. chee825 - Winter 2004J. McLellan3 Dynamic Process Relationships dependence of current output on present and past values of –manipulated variable inputs –disturbance inputs process transfer function »“deterministic” trends between u and y disturbance component »relationship between (possibly stochastic) disturbance n and y

4. chee825 - Winter 2004J. McLellan4 Dynamic Models dependence on past values goal - estimate models of form example –how can we determine how many lagged inputs, outputs, disturbances to use? –correlation analysis - auto/cross-correlations

5. chee825 - Winter 2004J. McLellan5 Impulse Response processes have inertia »no instantaneous jumps »when perturbed, require time to reach steady state one characterization –impulse response –pulse at time zero enters the process

6. chee825 - Winter 2004J. McLellan6 Impulse Response Process Input u(k) output y(k) time impulse weights h(k), k=0,1,2,...

7. chee825 - Winter 2004J. McLellan7 Impulse Response as a Weighting Pattern Given sequence of inputs, we can predict process output impulse response infinitely long if process returns to steady state asymptotically

8. chee825 - Winter 2004J. McLellan8 Interpretation Sum of impulse contributions 0 impact of input move 1 time step ago impact of input move 2 time steps ago output y(k) time (ZOH

9. chee825 - Winter 2004J. McLellan9 Impulse Response Model impulse response is an example of non- parametric model »practically - truncate and use finite impulse response (FIR) form impulse response model can be considered in –control modeling »model predictive control (e.g., DMC) –disturbance modeling »time series -- moving average representation

10. chee825 - Winter 2004J. McLellan10 Disturbance Models in Impulse Response Form inputs are random “shocks” »white noise fluctuations - random pulses impulse response weights describe how fluctuations in past affect present measurement white noise pulse impulse response parameters

11. chee825 - Winter 2004J. McLellan11 Disturbance Models in Impulse Response Form also referred to as a moving average representation –moving average of present and past random shocks entering process

12. chee825 - Winter 2004J. McLellan12 Difference Equation Models recursive definition describing dependence of current output on previous inputs and outputs y - output; u - manipulated variable input; e - random shocks (white noise) example - ARMAX(1,1,1) model with time delay of 1

13. chee825 - Winter 2004J. McLellan13 The Backshift Operator dynamic models represent dependence on past values - need a method to represent “lag” backshift operator q -1 : forward shift -- using q: alternate notations -- B, z -1 »z -1 - used in discrete control as argument for Z-transform

14. chee825 - Winter 2004J. McLellan14 Transfer Function Models start with difference equation model and introduce backshift operators relative to current time “t” “solve” for y(t) in terms of u(t) and e(t) process transfer function disturbance transfer function

15. chee825 - Winter 2004J. McLellan15 Transfer Function Models General form - ratios of polynomials in q -1 Roots of denominator represent poles »of process input-output relationship »of disturbance input-output relationship Roots of numerator represent zeros

16. chee825 - Winter 2004J. McLellan16 A Stability Test continuous control - poles must have negative real part in Laplace domain (complex plane) discrete dynamics? Consider the sum… if Geometric Series

17. chee825 - Winter 2004J. McLellan17 Stability Test Now consider if. Impulse response of is {1,a,a 2,…} which is stable if Root of denominator is q=a, or q -1 =a -1

18. chee825 - Winter 2004J. McLellan18 Stability Test Dynamic element is STABLE if »root in “q” is less than 1 in magnitude »root in “q -1 ” is greater than 1 in magnitude Approach - check roots of denominator »based on argument that higher order denominator can be factored into sum of first-order terms - Partial Fraction Expansion »each first-order term corresponds to a elementary response - decaying or exploding

19. chee825 - Winter 2004J. McLellan19 Moving Between Representations From the preceding argument, so

20. chee825 - Winter 2004J. McLellan20 Moving Between Representations transfer function impulse response The transformation can be achieved by solving for the impulse response of the discrete transfer function, or by “long division”.

21. chee825 - Winter 2004J. McLellan21 Inversion We can express transfer fn. model as impulse response model - infinite sum of past inputs. Can we do the opposite? »express input as infinite sum of present and past outputs? »example as

22. chee825 - Winter 2004J. McLellan22 Invertibility Answer - this is the dual problem to stability, and is known as invertibility. We can invert the moving average term if -- »root in “q” is less than 1 in magnitude »root in “q -1 ” is greater than 1 in magnitude Invertibility corresponds to “minimum phase” in control systems, and is a “stability check” of the numerator in a transfer function.

23. chee825 - Winter 2004J. McLellan23 Invertibility One use: for some input u … Write as current input move past outputs (inertia of process)

24. chee825 - Winter 2004J. McLellan24 Invertibility Importance? »particularly in estimation, where we will use this to form residuals given model What are the values of a(t)’s? Reformulate white noise y(t) ’s - measured quantities

25. chee825 - Winter 2004J. McLellan25 Representing Time Delays Using the backshift operator, a delay of “f” steps corresponds to: Notes -- »f is at least one for sampled systems because of sampling and “zero-order hold” »effect of current control move won’t be seen until at least the next sampling time