Process Dynamics & Control LECTURE 1: INTRODUCTION OF MODEL PREDICTIVE CONTROL A Multivariable Control Technique for the Process Industry

Transcription

1 Process Dynamics & Control LECTURE 1: INTRODUCTION OF MODEL PREDICTIVE CONTROL A Multivariable Control Technique for the Process Industry Jong Min Lee Chemical and Biological Engineering Seoul National University

2 What is MPC? Process PID Actuator or Gp Low-level Loops Sensor Up-to-date Process Information Optimal Process Adjustment Simulation/Optimization Package Database JM Lee Computer Dynamic Process Model Objective & Constraints Connection to Information System 2

3 Main Algorithm Past measurements Future y max target projected output u max future input t t+1 t+m-1 t+p input Horizon JM Lee

4 Some Key Features Computer based: sampled-data control Model based: requires a dynamic process model Predictive: makes explicit prediction of the future time behaviour of CVs within a chosen window Optimization based: performs optimization (numerical search) online for optimal control adjustments No explicit form of control law just model, objective, and constraints are specified Integrated: constraint handling and economic optimization with regulatory and servo control Receding Horizon Control: repeats the prediction and optimization at each sample time step to update the optimal input trajectory after a feedback update JM Lee

5 Exemplary Algorithm u Plant y Receding Horizon Control only the adjustment for the current sample time is implemented and the rest are re-optimized at the next sample time step after a new feedback update reference Optimizer t=k Observer ˆX reference Y t ( ) =f ˆXt, U t ( ) U t ( ) Z t+p Control Objective min U t ( ) t `1 [Error( )] + `2 [Input( )] d U( ) 2 U, Constraints Y t ( ) 2 Y JM Lee

6 Analogy JM Lee 45804

7 Industrial Use of MPC Initiated at Shell Oil and other refineries during late 0s and early 80s Various commercial software DMCplus Aspen Tech RMPCT Honeywell Dozen+ other players (eg, 3DMPC-ABB) > 3000 worldwide installations Predominantly in the oil and petrochemical industries but the range of applications is expanding Models used are predominantly empirical models developed through plant testing Technology is used not only for multivariable control, but for most economic operation within constraint boundaries JM Lee 45804

8 Survey Result (1) Applications by 5 major MPC vendors in North America / Europe (Badgwell and Qin, 2003) JM Lee

9 Survey Result (2) - Japan (Oshima, 1995) JM Lee

10 Reason for Popularity (1) MPC provides a systematic, consistent, and integrated solution to process control problems with complex features: - Delays, inverse responses and other complex dynamics - Strong interactions (eg, large RGA) - Constraints (eg, actuator limits, output limits) Supervisory Control Selectors, Switches, Delay Compensations, Antiwindups, Decouplers, etc More and more optimization is the MPC level Process Optimization Advanced Multi-Variable Control MPC Low-level PID Loops Low-level PID Loops JM Lee

11 Example 1: Blending System Control u 1 u 2 u 3 Valve Positions Stock Additive A Additive B Blending System Model r A = Additive A stock r B = Additive B stock q total blend flow Control r A and r B Control q if possible Flowrates of additives are limited JM Lee

12 Classical Solution Valveposition controller 95% Setpoint VPC Selector < Setpoint FC Feedback Stock FT FT Additive A X Ratio setpoint FT FC > High selector Blend of A and B X FC Additive B Ratio setpoint FT JM Lee

13 MPC: each time k p: size of prediction window min u 1 (j),u 2 (j),u 3 (j) j = k,, k + p 1 px (r A (k + i k) ra) 2 +(r B (k + i k) rb) 2 + (q(k + i k) q ) 2 i=1 (u i ) min apple u i (j) apple (u i ) max, i =1,,3, << 1 JM Lee

14 Advantages of MPC over Traditional APC Integrated solution Automatic constraint handling Feedforward/feedback No need for decoupling or delay compensation Efficient utilization of degrees of freedom Can handle nonsquare systems (eg, more MVs and CVs) Assignable priorities, ideal settling values for MVs Consistent, systematic methodology Realized benefits Higher online times Cheaper implementation Easier maintenance JM Lee

15 Reason for Popularity (2) Emerging popularity of online optimization Process optimization and control are often conflicting objectives Optimization pushes the process to the boundary of constraints Quality control determines how close one can push the process to the boundary Implications for process control High performance control is needed to realize online optimization Constraint handling is a must The appropriate tradeoff between optimization and control is timevarying and is best handled within a single framework Model Predictive Control JM Lee

16 Bi-Level Optimization Used in MPC Steady-State Optimization (LP) Economics based Objective (Maximum profit or thruput, minimum utility) Control Based Constraints Optimal setting values for the inputs and outputs (setpoints) Adjustments to setpoints of low level loops or control valves Dynamic Optimization (QP) Steady-state prediction model New measurements (Feedback update) Minimization of Error (= setpt output and input) Constraints on actuator limits and safety-sensitive variables JM Lee

17 New Operational Hierarchy and Role of MPC Customer Strategic Planning $ month ~ year Production Planning Plant Scheduling Real Time Optimizer Model Predictive Control Low-Level Control $ week ~ month $ min ~ day sec $ Move the plant to the current optimal condition fast and smoothly w/o violating constraints: Local optimization + control JM Lee

18 An Exemplary Application: Ethylene Plant Primary Quench Fractionator Tower Demethanizer Deethanizer Ethylene Fractionator Hydrogen Furnaces Methane Ethylene Charge Gas Compressor Chilling Ethane Propylene Propane Feedstock Naphtha Light H-C Fuel Oil Depropanizer B - B Gasoline Propylene Fractionator Debutanizer JM Lee

19 Importance of Modeling/Sys-ID Model is the most critical element of MPC that varies the most from application to application Almost all models used in MPC are typically empirical models identified through plant tests rather than first-principles models Step responses, pulse responses from plant tests Transfer function models fitted to plant test data Up to 80% of time and expense involved in designing and installing a MPC is attributed to modeling/system identification Keep in mind that obtained models are imperfect (both in terms of structure and parameters) Importance of feedback update to correct model prediction or model parameters/states Penalize excessive input movements JM Lee

20 Design Effort for Two Approaches Process Analysis Design and Tuning of Controller Modeling & Identification Control Specification Traditional Control MPC JM Lee

21 Challenges for MPC Efficient identification of control-relevant model Managing the sometimes exorbitant online computational load Nonlinear models è Nonlinear Programs (NLP) Hybrid system models (continuous dynamics + discrete events or switches, eg, pressure swing adsorption) è Mixed Integer Programs (MIP) Difficult to solve these reliably online for large-scale problems How do we design model, estimator (of model parameters and state), and optimization algorithm as an integrated system that are simultaneously optimized, rather than as disparate components? Long-term maintenance of control system JM Lee

22 Control Relevant Modeling Coupling between Modeling and Control Model Quality (Error or Uncertainty ) System ID Test signal characteristics Model structure Data filtering Parameter fitting Model MPC Design Choice of objective function and constraints Choice of horizon sizes Choice of online estimator Sensitivity of Control Performance to Model Errors JM Lee

23 Iterative Model/Controller Refinement Identification & Controller Design Closed-loop Data New Controller Gc Gc Gp Closed-loop Operation and Testing JM Lee

24 Comparison of Computational Load Classical Optimal Control MPC Offline Analysis and Computation Explicit Control Law u = f(x) Online Computation Model; Obj Fcn; Constraints min f(x) g(x) > 0 Offline Computation Online Computation (Estimation, Prediction, & Optimization) Limited by the ability to derive the explicit control law analytically or with reasonable offline computation JM Lee Limited by available online computational power and numerical methods to solve online optimization reliably 24

25 Coupling between Online Estimation and Control Calculation Modeling (System ID) Quality of Information for Estimation Online Estimation of State & Model Parameters Model w/ parameters & states to estimate Uncertainty in Prediction = Risk Prediction Real-Time Adjustment Online Optimal Control Calculation JM Lee

26 Integrated MPC, Performance Monitoring, and Closed-Loop Identification Adjustments MPC Online Model Identification Measurements Detection and Diagnosis of Abnormal Situation Operation shifts: model parameter changes Abnormal disturbances (size & pattern) Instrumentation/Equipment Faults, Poisoning, etc Process Monitoring JM Lee

27 Conclusion MPC is the established advanced multivariable control technique for the process industry It is already an indispensable tool and its importance is continuing to grow It can be formulated to perform some economic optimization and can also be interfaced with a larger-scale (eg, plant-wide) optimization scheme Obtaining an accurate model and having reliable sensors for key parameters are key bottlenecks A number of challenges remain to improve its use and performance JM Lee

28 45804 Process Dynamics & Control Lecture3: Dynamic Matrix Control (DMC) Jong Min Lee School of Chemical and Biological Engineering Seoul National University 1

29 In this lecture, we will discuss Process representation: step response model Prediction (perfect model) Incorporation of feedback Optimization: unconstrained and constrained QP Implementation 2

30 Dynamic Matrix Control First appeared in the open literature in 199 (Cutler and Ramaker; Prett and Gillette) - with notable success on several Shell processes for many years Reformulation as a quadratic program by Garcia and Morshedi in Quadratic Dynamic Matrix Control AspenTech: DMCplus Prototype of commercial algorithms presently used in the process industry 3

31 Process representation Stable, SISO: u y S n = S n+1 = S n+2 = = S S 0 = 0 1 Dynamic System S n S n+1 S 2 T s S 0 S 1 T s unit step-response function: S =[S 1, S 2, S 3,, S n ] T Complete description of the process requires n step response coefficients 4

32 Principle of superposition u u u u 2 1 1/2 = u(0) u(1) u(2) -3/2 y y(3) y(4) y(1) = y(0) + S 1 u(0) y(2) = y(0) + S 1 u(1) + S 2 u(0) y(2) y(5) = y(0) y(1) y(k + 1) = y(0) + nx 1 i=1 S i u(k i + 1) + S n { u(k n + 1) + u(k n)+ u(0)}? = y(0) + nx 1 i=1 S i u(k i + 1) + S n u(k n + 1) 5 u(k i + 1) = u(k i + 1) u(k i)

33 Elements of DMC Past Future ~ y(k) ~ y(k+1) ~ y(k+2) target u(k) u(k+1) k k+1 k+m-1 k+p input Horizon

34 Predictions

35 1 Prediction (stable, SISO) At time k: we know y(k) and need to compute u(k), which we don t know yet ŷ(k + 1): prediction of y(k+1) made at time k ŷ(k + 1) = Assume y(0) = 0 nx 1 i=1 ŷ(k + 1) = S 1 Effect of current control action S i u(k i + 1) + S n u(k n + 1) u(k)+ nx 1 i=2 S i u(k i + 1) + S n u(k n + 1) Effect of past control actions Substitute k = k+1, and ŷ(k + 2) = S 1 u(k + 1) + S 2 u(k)+ Effect of future control action Effect of current control action nx 1 i=3 S i u(k i + 2) + S n u(k n + 2) Effect of past control actions 8

36 j-step ahead prediction ŷ(k + j) = jx S i u(k + j i)+ i=1 Effect of current and future control actions nx 1 i=j+1 S i u(k + j i)+s n u(k + j n) Effect of past control actions Let ŷ 0 (k + j) = nx 1 i=j+1 S i u(k + j i)+s n u(k + j n) This is referred to as predicted unforced response with past inputs only U =[, u(k 2), u(k 1), 0, 0, 0, ] T for j = 1 ŷ(k + j) = jx S i u(k + j i)+ŷ 0 (k + j) i=1 9

37 Multiple predictions Ŷ(k + 1) =[ŷ(k + 1), ŷ(k + 1),, ŷ(k + p)] T Ŷ 0 (k + 1) = ŷ 0 (k + 1), ŷ 0 (k + 2),, ŷ 0 (k + p) T U(k) =[ u(k), u(k + 1),, u(k + m 1)] T p: prediction horizon, m: control horizon m apple p apple n + m In a matrix form: Ŷ(k + 1) = S U(k)+Ŷ0 (k + 1) S = 2 4 S S 2 S S 3 S 2 S 1 0 S m S m 1 S m 2 S 1 S m+1 S m S m 1 S 2 S p S p 1 S p 2 S p m

38 Output feedback and bias correction So far, we have not utilized the latest observation, y(k) The fact is that there is no perfect model Corrected prediction by adding a constant bias term ỹ(k + j) =ŷ(k + j)+b(k + j) b(k + j) =y(k) ŷ(k) ŷ(k): one-step ahead prediction made at the previous time instance, k-1 ỹ(k + j) =ŷ(k + j)+[y(k) ŷ(k)] Ỹ(k + 1) = S U(k)+Ŷ0 (k + 1) + [y(k) ŷ(k)] 1 Ỹ(k + 1) = [ỹ(k + 1), ỹ(k + 2),, ỹ(k + p)] T 1 =[1, 1,,, 1] T 11

39 Recursive update of unforced response For stable models, one can update the predicted unforced response after u(k) is computed works like a state; hence you need n not p or 2 4 Ŷ 0 n(k + 1) = MŶ0 n(k)+s u(k) ŷ 0 (k + 1) ŷ 0 (k + 2) 5 = ŷ 0 (k) S 1 0 ŷ 0 (k + 1) S 2 4 ŷ 0 4 (k + p) ŷ 0 (k + p 1) S p n n n 3 5 u(k) 12

40 Why?? To achieve good control performance - Ỹ(k + 1) should be close to the true open-loop output - This requires that n, the number of coefficient matrices in S* is chosen that Sn = Sn+1 (ie, plant should be stable), otherwise MŶ0 will be in error It also requires the feedback term stays approximately constant (step disturbance) 13

41 1 Prediction (stable, MIMO) 14

42 2-by-2 system ŷ 1 (k + 1) = nx 1 i=1 nx 1 + S 11,i u 1 (k i + 1) + S 11,n u 1 (k n + 1) i=1 S 12,i u 2 (k i + 1) + S 12,n u 2 (k n + 1) ŷ 2 (k + 1) = nx 1 i=1 nx 1 + S 21,i u 1 (k i + 1) + S 21,n u 1 (k n + 1) i=1 S 22,i u 2 (k i + 1) + S 22,n u 2 (k n + 1) 15

43 Vector notation y = Ỹ(k + 1) = S = y 1 y 2 y m u = 4 ỹ(k + 1) ỹ(k + 2) ỹ(k + p) 3 5 u 1 u 2 u r 3 5 S mp-by-1 Ŷ 0 (k + 1) = S 2 S S m S m 1 S 1 S m+1 S m S 2 S p S p 1 S p m+1 Ỹ(k + 1) = S U(k)+Ŷ0 (k + 1) + I p [y(k) ŷ(k)] ŷ 0 (k + 1) ŷ 0 (k + 2) ŷ 0 (k + p) S i = 3 5 Use up to p only out of n 2 4 U(k) = I p = u(k) u(k + 1) u(k + m 1) S 11,i S 12,i S 1r,i S 21,i S 2r,i S m1,i S mr,i I I 3 5 pny-by-ny

44 Recursive update of unforced response Ŷ 0 n(k + 1) = MŶ0 n(k)+s u(k) 2 4 or ŷ 0 (k + 1) ŷ 0 (k + 2) ŷ 0 (k + p) I m = 0 0 I ŷ 0 (k) S 1 m 0 ŷ 0 (k + 1) S I m 5 ŷ 0 (k + p 1) S p I n m n n 3 5 u(k) 1

45 Control calculations 18

46 Objective function At time k, minimize the predicted deviation of the output from the setpoint with some penalty on the input movement size measured in terms of the quadratic norm min U(k) ( px (y r (k + i) ỹ(k + i)) T Q (y r (k + i) ỹ(k + i)) + i=1 mx 1 `=0 u T (k + `)R u(k + `) ) Q, R : weighting matrices (diagonal) 19

47 Constraints Past measurements Future future input y max target projected output u max Input magnitude u min apple u(k + `) apple u max Input rate u(k + `) apple u max t t+1 t+m-1 t+p Output magnitude input Horizon y min apple ỹ(k + i) apple y min 20

48 Solve: quadratic program 1 min U(k) 2 UT (k)h U(k)+f T U(k) A U(k) apple b H f A b U(k) : Hessian matrix : gradient vector : constraint matrix : constraint vector : decision variable We need to convert the MPC objective and constraints to the standard QP form 21

49 Unconstrained problem 1 min U(k) 2 UT (k)h U(k)+f T U(k) Take the gradient wrt the input: H U(k)+f =0 U(k) = H 1 f 22

50 Objective function in quadratic form min U(k) ( px (y r (k + i) ỹ(k + i)) T Q (y r (k + i) ỹ(k + i)) + i=1 mx 1 `=0 u T (k + `)R u(k + `) ) 2 4 y r (k + 1) ỹ(k + 1) y r (k + 2) ỹ(k + 2) y r (k + p) ỹ(k + p) 3 T Q Q Q y r (k + 1) ỹ(k + 1) y r (k + 2) ỹ(k + 2) y r (k + p) ỹ(k + p) u(k) u(k + 1) u(k + m 1) 3 T R R R u(k) u(k + 1) u(k + m 1) 3 5 T Y r (k + 1) Ỹ(k + 1) Q Y r (k + 1) Ỹ(k + 1) + U T (k) R U(k) 23

51 Not done yet! T Y r (k + 1) Ỹ(k + 1) Q Y r (k + 1) Ỹ(k + 1) + U T (k) R U(k) This yields Ỹ(k + 1) = S U(k)+Ŷ0 (k + 1) + I p [y(k) ŷ(k)] " T (k + 1) Q"(k + 1) 2" T (k + 1) QS U(k)+ U T (k) S T QS + R U(k) where "(k + 1) = Y r (k + 1) Ŷ 0 (k + 1) I p [y(k) ŷ(k)] is a known term Hessian (a constant matrix): H = S T QS + R gradient vector (must be updated at each time): f T = " T (k + 1) QS 24

52 Constraints in linear inequality form u min apple u(k + `) apple u max u(k + `) apple u max A U(k) apple b y min apple ỹ(k + i) apple y min i =1,,p ` =0,,m 1 25

53 Input magnitude constraint u min apple u(k + `) apple u max, ` =0,, m 1 u(k 1) `X i=0 u(k + i) apple u min u(k 1) + `X i=0 u(k + i) apple u max I 0 0 I I 0 0 I I I I 0 0 I I 0 0 I I I u(k) u(k + 1) u(k + m 1) 3 5 apple u min u(k 1) u min u(k 1) u min u(k 1) u max u(k 1) u max u(k 1) u max u(k 1) I L 2

54 Input rate constraints u(k + `) apple u max ` =0,,m 1 u max apple u(k + `) apple u max u(k + `) apple u max u(k + `) apple u max I I I I I I u(k) u(k + 1) u(k + m 1) 3 5 apple u max u max u max u max u max u max I 2

55 Output magnitude constraints y min apple ỹ(k + i) apple y max, i =1,,p ỹ(k + i) apple y max ỹ(k + i) apple y min apple S U(k)+Ŷ0 (k + 1) + I p (y(k) ŷ(k)) S U(k) Ŷ 0 (k + 1) I p (y(k) ŷ(k)) 2 3 y max y max Y max = Y min = apple apple y min y min Y max Y min 3 5 y max y min apple S S U(k) apple apple Y max Ŷ 0 (k + 1) I p (y(k) ŷ(k)) Y min + Ŷ0 (k + 1) + I p (y(k) ŷ(k)) 28

56 In summary, 2 4 I L I L I I S S 3 5 U(k) apple u min u(k 1) u min u(k 1) u max u(k 1) u max u(k 1) u max u max u max u max 3 5 apple Y max Ŷ 0 (k + 1) I p (y(k) ŷ(k)) Y min + Ŷ0 (k + 1) + I p (y(k) ŷ(k)) 3 5 A U(k) apple b 29

57 Solving QP Quadratic program: minimization of quadratic function subject to linear inequality constraints QPs are convex and therefore fundamentally tractable Off-the-shelf solvers (eg, QPSOL, QUADPROG) are available but further customization is desirable (to exploit the structure in the Hessian and constraint matrices) Complexity of a QP is a complex function of the dimension/structure of Hessian, as well as the number of constraints 30

58 Active set method Interior point method - Barrier function 31

59 Real-time implementation 1 Initialization: Initialize the memory vector and the reference vector Ŷ(0) and the reference vector Set k = 0 2 Memory update: Ŷ 0 (k + 1) = MŶ0 (k)+s u(k) 3 Reference vector update 4 Measurement intake: Take in new measurement y(k) and d(k) 5 Calculation of the gradient vector and constraint vector Solve QP Implementation of input u(k) =u(k 1) + u(k) 8Go back to step 2 after setting 32

60 45804 Process Dynamics & Control Lecture 4: Sampling and Representation of Sampled Signals Jong Min Lee Chemical & Biomolecular Engineering Seoul National University April 1, / 1

61 Overview ZOH u u(t) y(t) k y k D/A System A/D A computer oriented mathematical model (or discrete-time model) relates u k to y k does not give information on intersample behaviour can be described using difference equation or a pulse transfer function 2 / 1

62 Input-Output Model Input-output model describes a relationship between input u k and output y k Generally, it takes the form of the following difference equation: y k = a 1 y k 1 a n y k n + b 1 u k b m u k m With some abuse of notation, the above is written as (1 + a 1 q a n q n )y k = (b 1 q b m q m )u k z-transform Y(z) = b 1z 1 +b 2 z 2 + b m z m U(z) 1+a 1 z 1 + a nz n The order of the transfer function is determined by max(n, m) Denominator: Autoregressive Terms Numerator: Moving Average Terms 3 / 1

63 Discrete-Time Pole Consider the first-order system y k = ay k 1 + u k 1 Y(z) U(z) = z 1 1 az 1 One can expand the above as a power series of z 1 around z 1 = 0: Y(z) U(z) = 1 z + a z 2 + a2 z an 1 z N + truncation error Obvious convergence (stability) condition is a < 1 Note that a is the pole of Y(z) U(z) 4 / 1

64 State-Space Model A model can also be given in terms of the following matrix difference equation: x k+1 = Φx k + Γu k y k = Cx k + Du k x k is called a state vector and stores the effect of past input (u k 1, u k 2, )on the current and future output The state variables may or may not have physical meanings An equivalent input-output representation can easily be derived by performing the z-transform to the above: zx(z) = Φx(z) + Γu(z) Y(z) = Cx(z) + Du(z) Y(z) U(z) = C(zI Φ) 1 Γ + D 5 / 1

65 Input-Output Models from Discretization of Continuous TF (Optional) u k ZOH D/A G(s) A/D y k G(z) z-transform describes a discrete signal as ``impulse train" when viewed in continuous time A zero-order hold converts the sampled signal to a piece-wise constant signal (train of pulses) Hence, we need to derive a pulse transfer function for zero-order hold The basic idea is Y(z) {L {G(s) U(z) = G(z) = Z 1 1 }} e sh s / 1

66 Input-Output Models from Identification Suppose one is interested in fitting an n th -order transfer function model In the time domain, this corresponds to Y(z) U(z) = b 1z 1 + b 2 z b m z m 1 + a 1 z 1 + a 2 z a n z n y(k) = a 1 y(k 1) a 2 y(k 2) a n y(k n) + b 1 u(k 1) + b 2 u(k 2) + + b m u(k m) Notice that there is at least one time-delay between the input and output due to the presence of ZOH element The above is a linear regression model y(k) = ϕ T (k)θ where ϕ T (k) = [ y(k 1) y(k n) u(k 1) u(k m)], θ T = [a 1 a n b 1 b n] / 1

67 Estimating θ from N sample data points Given N input-output samples, Solution to the Least Squares problem ˆθ N = Example: Given the transfer function: ( Φ T NΦ N ) 1 Φ T N Y N where Φ N = G(z) = b 1z 1 + b 3 z a 1 z 1 + a 2 z 2 ϕ T (1) ϕ T (N) write the difference equation corresponding to G(z) and also form the Φ matrix suitable to estimate the parameters using the LS method 8 / 1

68 Linear Regression Solution y(k) + a 1y(k 1) + a 2y(k 2) = b 1u(k 1) + b 3u(k 3) The Φ N matrix corresponding to any difference equation is formed by first forming the Y N matrix, which is constructed by looking at the term with the largest sample distance between y(k) and any of y(k n) and u(k m) In the above example this term is u(k 3) This is done to avoid negative sample indices while writing Φ matrix Thus, y(4) y(3) y(2) u(3) u(1) y(5) Y N = = Φ y(4) y(3) u(4) u(2) N = y(n) y(n 1) y(n 2) u(n 1) u(n 3) 9 / 1

69 State-Space Models from Discretization Suppose we are given a model described by a system of linear differential equation: dx dt = Ax + Bu y = Cx + Du In the above, x is an n-dimensional vector Suppose that (1) a zero-order hold is used and (2) sampling is synchronized for all inputs and outputs Then, treating t = kh as the initial time and x k = x(kh) as an initial condition we have x(t) = e A(t kh) x k + t kh e A(t τ) Bu(τ)dτ 10 / 1

70 Discretization of Continuous SS Model Evaluating the above at t = kh + h with the fact that u(t) = u k for kh t < kh + h (due to the zero-order-hold assumption), we obtain x k+1 = e A(kh+h kh) x k + kh+h kh e A(kh+h τ) Bu(τ)dτ ( h ) = e Ah x k + e As ds Bu k [s = kh + h τ] 0 Now we can write the propagation of variables from one sample time to next as x k+1 = Φx k + Γu k y k = Cx k + Du k where Φ = e Ah Γ = h 0 e As dsb 11 / 1

71 Delays Can Be Easily Incorporated into the Discrete Model Case I: 0 < θ h Recall dx = Ax + Bu(t θ) dt Note that x(kh + h) = e Ah x(kh) + kh+h kh e A(kh+h τ) Bu(τ θ)dτ { uk kh + θ τ < kh + h u(τ θ) = kh τ < kh + θ u k 1 Substituting the above and making the change of variable s = kh + h τ θ ) ( h θ ) x k+1 = e Ah x k + (e A(h θ) e As dsb u k 1 + e As dsb u k 0 0 = Φx k + Γ 1 u k 1 + Γ 0 u k 12 / 1

72 Discrete SS Model with Delays (Cont'd) We can put the above in the standard form as follows: [ ] [ ] [ ] [ xk+1 Φ Γ1 x = k Γ0 + u k 0 0 u k 1 I ] u k y k = Cx k + Du k = [ C 0 ] [ x k u k 1 ] + Duk Hence the state vector at the k th time consists of x k and u k 1 This makes sense since when we have delay ( h), the effect of u k 1 has not been fully stored in x k Case II: θ = (d 1)h + θ where 0 < θ h and d 1 Note that for d = 1, we have the previous case As before x(kh + h) = e Ah x(kh) + kh+h kh e A(kh+h τ) Bu(τ θ)dτ 13 / 1

73 But this time Hence, u(τ θ) = { uk d+1 kh + θ τ < kh + h u k d kh τ < kh + θ x k+1 = e Ah x k + ( θ e A(h θ ) ( ) h θ + e As dsb 0 0 e As dsb u k d+1 = Φx k + Γ 1 u k d + Γ 0 u k d+1 ) u k d 14 / 1

74 We can put the above in the standard form as follows: x k+1 u k d+1 u k 1 u k = Φ Γ 1 Γ I 0 0 I 0 0 x k u k d u k 2 u k I u k y k = [ C ] x k u k d u k 2 u k 1 + Du k Note that the state vector at the k th time must include u k 1,, u k d since the effect of past d inputs has not been stored in x k 15 / 1

75 State-Space Models from Identification One can also obtain a discrete state-space model from data This can be done by Using methods called subspace ID that directly gives a model in the discrete state-space form Identifying a transfer function model and then performing a ``realization" on it (which means finding an I/O-wise equivalent state-space model representation) 1 / 1

76 45804 Process Dynamics & Control Lecture 5: System Identification: Introduction Jong Min Lee Chemical & Biomolecular Engineering Seoul National University April 20, / 1

77 References L Ljung, System Identification: Theory for the User, Prentice Hall Soderstrom, T and P Stoica, System Identification, Prentice Hall Box, G E P and G M Jenkins, Time Series Analysis: Forecasting and Control, Holden-Day, / 1

78 First-Principles Modeling Usually involves fewer measurements; requires experimentation only for the estimation of unknown parameters Provides information about internal state of the process Promotes fundamental understanding of the internal workings of the process Requires fairly accurate and complete process knowledge Not useful for poorly understood and/or complex processes Naturally produces both linear and nonlinear models System Identification Requires extensive measurements Provides information only about the portion of the process Treats the process like a ``black box" Requires no such detailed knowledge Quite often proves to be the only alternative for poorly understood/complex processes Requires special methods for nonlinear models 3 / 1

79 Objective of Sys ID From I/O Data Set: {y(k), u(k), k = 1,, N} Identify ỵ(k) = G(q)u(k) + H(q)ε(k) where or G(q): plant transfer function (deterministic part) H(q): disturbance transfer function (stochastic, noise part) ε: white noise x(k + 1) = Ax(k) + Bu(k) + Kε(k) y(k) = Cx(k) + ε(k) System identification at a more general level includes other tasks such as data generation, data pretreatment, and model validation 4 / 1

80 Plant vs Noise Model u(k) Plant Model G(q) y(k) According to the figure above, the output can be exactly calculated once the input is known In most cases, this is unrealistic There are always signals beyond our control that also affect the system Assume that such effects can be lumped into an additive term w(k) at the output y(k) = g(τ)u(k τ) + w(k) τ=1 Note: g(τ) is the impulse response, which is obtained by the unit pulse input 5 / 1

81 Then, we have w(k) u(k) Plant Model + G(q) y(k) The value of disturbance (w(k)) is not known beforehand So, we employ a probabilistic framework to describe future disturbances We assume that w(k) is driven by a white noise sequence ε(k) for simplicity / 1

82 ε(k) Disturbance Model H(q) u(k) Plant Model + G(q) y(k) ε(k): white noise / 1

83 Parametric vs Nonparametric Methods 1 Parametric Methods Select the best one among a confined set of possible models Finite dimensional parameters Ex) transfer function (matrix) of given order, ``Finite" impulse response identification 2 Nonparametric Methods Time domain: Step response, Impulse response, Correlation analysis Frequency domain: Fourier analysis, Spectral analysis End Objective: Obtain a model providing a good (multi-step) prediction with the intended feedback control loop in place 8 / 1

84 Model Structure for Parametric Identification Standard Form (SISO): y(k) = G(q, θ)u(k) + H(q, θ)ε(k) ε(k): a white noise sequence H(q): stable and stable invertible transfer function Differenced Form: If the process mean shifts continuously or time to time use y(k) = G(q, θ) u(k) + H(q, θ)ε(k) 9 / 1

85 ARX (Auto Regressive exogenous) y(k) + a 1 y(k 1) + a n y(k n) = b 1 u(k 1) + + b m u(k m) + ε(k) G(q, θ) = B(q) A(q) 1 H(q, θ) = A(q) = b 1q b m q m 1 + a 1 q a n q n = a 1 q a n q n For ARX structure, use of a very high order model is often necessary 10 / 1

86 ARMAX (Auto Regressive Moving Average exogenous) y(k) + a 1 y(k 1) + + a n y(k n) = b 1 u(k 1) + + b m u(k m) + ε(k) + c 1 ε(k 1) + + c l ε(k l) H(q) = C(q) A(q) = 1 + c 1q c l q l 1 + a 1 q a n q n 11 / 1

87 OE (Output Error) Structure ỹ(k) + a 1 ỹ(k 1) + + a n ỹ(k n) = b 1 u(k 1) + + b m u(k m) ỹ: deterministic output G(q, θ) = A(q) B(q) y(k) = ỹ(k) + ε(k) and H(q) = 1 The OE structure can also encompass the case where the noise model is set a priori: y(k) = G(q, θ)u(k) + H(q)ε(k) 1 H(q) y(k) = G(q, θ) 1 u(k) + ε(k) H(q) 12 / 1

88 Orthogonal Expansion Model A special kind of OE structure where G(q) = n b i B i (q) i=1 where B i (q) are orthogonal basis functions For example, B i (q) = q i Finite Impulse Response model B i (q) = ( ) i 1 1 α 2 1 αq q α q α Laguere model 13 / 1

89 Other Structures Box-Jenkins structure: y(k) = B(q) C(q) u(k) + A(q) D(q) ε(k) ARIMAX structure (Auto Regressive Integrator Moving Average exogenous) y(k) = B(q) A(q) u(k) q 1 C(q) A(q) ε(k) 14 / 1

90 Nonparametric Model: Impulse Response u u = {1, 0, 0, } y = {0, H 1, H 2,, H n, H n+1, 0, } y 1 h Time H 1 H 2 H 3 Time 15 / 1

91 Nonparametric Model: Step Response u u = {1, 1, 1, } y = {0, S 1, S 2, S 3, } y 1 h Time S 1 S 2 S 3 Time 1 / 1

92 Major Steps Gathering of data through a plant test Data conditioning and pretreatment Transition of data to a model: model structure selection and parameterization plus parameter estimation Validation 1 / 1

93 The System Identification Loop Experiment Design Prior Knowledge Data Choose Model Set Choose Criterion of Fit Calculate Model Validate Model Not OK: Revise OK: Use It 18 / 1

94 Step Testing Procedure 1 Assume operation at steady-state with controlled var (CV): y(t) = y 0 for t < t 0 manipulated var (MV): u(t) = u 0 for t < t 0 2 Make a step change in u of a specified magnitude, u for 3 Measure y(t) at regular intervals: u(t) = u 0 + u for t t 0 y(k) = y(t 0 + kh) for k = 1, 2,, N where h: the sampling interval Nh: is approximate time required to reach steady state 4 Calculate the step response coefficients from the data S(k) = y(k) y 0 u for k = 1, 2,, N 19 / 1

95 Discussions 1 Choice of sampling period h For modelling, best h is one such that N = Ex: If G(s) = Ke θs, then settling time 4τ + θ τs + 1 Therefore, h 4τ + θ N = 4τ + θ = 01τ θ 40 May be adjusted depending on control/operation objectives 2 Choice of Step Size ( u) Too small: May not produce enough output change Low signal to noise ratio Too big: Shift the process to an undesirable condition Nonlinearity may be induced Trial and error is needed to determine the optimum step size 20 / 1

96 Discussions on Step Testing (Cont'd) 3 Choice of number of experiments Averaging results of multiple experiments reduces impact of disturbances on calculated S(k)'s Multiple experiments can be used to check model accuracy by cross-validation (Data sets for Identification Data set for Validation) 4 An appropriate method to detect steady state is required 5 While the steady state (low frequency) characteristics are accurately identified, high frequency dynamics may be inaccurately characterized 21 / 1

97 Procedure for Pulse Testing (Impulse Response) 1 Steady operation at y 0 and u 0 2 Send a pulse of size δu lasting for 1 sampling period 3 Calculate pulse response coefficients H(k) = y(k) y 0 δu for k = 1,, N 22 / 1

98 Discussions on Pulse Testing 1 Select h and N as for the step testing 2 Usually need δu u for adequate S/N ratio 3 Multiple experiments are recommended for the same reason as in the step testing 4 An appropriate method to detect steady state is required 5 Theoretically, pulse is a perfect (unbiased) excitation for linear systems 23 / 1

99 Input Design Why use test inputs other than a step or pulse? Pure step tests or pulse tests usually take too long and are impossible for some inputs More system excitation produces more information Completely random inputs (eg, RBS, PRBS) excite all frequencies with equal energy 24 / 1

100 u u Type of Inputs Random Binary Signal (RBS) or Pseudo-Random Binary Signal (PRBS) Random Noise Time Time 25 / 1

101 PRBS Size of u(t) is fixed and switches between two levels Choice of whether to switch or stay is random: flip a coin Sequence design choices are: Levels to switch between Base length of time between switch (period) Duration of experiment Trade-off between size of PRBS and duration of experiment Larger size and longer duration give better estimates Power of this signal is that you can do a small size (unnoticeable) for a long time to get a good result Base switching period Reflect process dynamics Set a ``dominant" time constant 2 / 1

102 PRBS: Distillation Column Example Time to steady-state is mins (τ = 10 15mins) Length of experiment ( hours) switches (not very many) Levels Reflux Rate Steam Rate Sequence Design: May want to start with a step of 3 4τ Choose start (+1 or -1 level) At next switch time flip a coin 2 / 1

103 Frequency Range of Input Excitation 1 Based on the step response, obtain τ p 2 Calculate the corner frequency ω CF = 1 τ p [rad/time] 3 Choose a sampling interval h based on earlier discussion 1 4 Nyquist frequency: 2h [cycles/time] = π h [rad/time] 5 Choose lower bound for the input frequencies as zero (in order to obtain a good estimate of the gain) Choose upper bound for the input frequencies as 25 3 ω CF ω N In MATLAB, u = idinput(2000, 'rbs', [0 001], [-1 1]); 28 / 1

104 Model Types and Transfer Function Model Types Output Error (least general) ARX ARMAX Box-Jenkins (most general) Process Transfer Function G p (q 1 ) = B(q) F(q) q (d+1) Zeros: roots of B(q) Poles: roots of F(q) Time Delay: d -- Note that extra 1 time-delay is naturally introduced by zero order hold and sampling, and d is pure time delay 29 / 1

105 Disturbance Modelling: Stochastic Processes Parametric Autoregressive (denominator) Moving average (numerator) AutoRegressive and Moving Average (ARMA) w(k) = C(q) D(q) ε(k) ARIMA (AutoRegressive Integrated Moving Average) Model w(k) = C(q) 1 D(q) (1 q 1 ) d ε(k) 30 / 1

106 Least Squares Identification Recall (from Lecture 5) that least square estimate of parameters is given as ˆθ N = ( ϕ T (1) Φ T ) 1 NΦ N Φ T N Y N where Φ N = ϕ T (N) for y(k) = ϕ T (k)θ where ϕ T (k) = [ y(k 1) y(k n) u(k 1) u(k m)], θ T = [a 1 a n b 1 b n ] 31 / 1

107 45804 Process Dynamics & Control Lecture b: Disturbance Modelling Jong Min Lee Chemical & Biomolecular Engineering Seoul National University May 9, / 21

108 Disturbance Modelling Why? Predict its effect on the output so that they can be eliminated Deterministic vs stochastic disturbances steps, pulses, sinusoids -- deterministic white noise, colored noise, integrated white noise, etc -- random Stochastic processes are convenient vehicles to describe them Stochastic disturbances and noise are almost always present Most disturbances, even deterministic ones, are unpredictable in terms of size, direction and time of occurrence 2 / 21

109 Linear Stochastic Models Important: In linear systems, it is not necessary to identify and model actual physical disturbance sources It is sufficient to model their overall effect on the output w: Physical disturbance variables or signals representing the collective effect of disturbances on the output Driven by white noise ε Transfer function model w(k) = H(q)ε(k) ε(k) Stochastic Model H(q) w(k) 3 / 21

110 y Linear Stochastic Models: Example Ambient temperature / pressure at Edmonton International Airport Power / water consumption in Edmonton Stock market Any ``unknown" or ``indescribable" disturbances of a process unit A stochastic process may look like gross trends random behaviour around trends Time 4 / 21

111 General Structure w(k) = moving average component {}}{ 1 + θ 1 q 1 + θ 2 q θ m q m 1 + ϕ 1 q 1 + ϕ 2 q ϕ n q n }{{} autoregressive component 1 (1 q 1 ) }{{ d ε(k) } integrating component Each part gives a different relationship between the current value of stochastic output (w(k)) with its past values (w(k 1), w(k 2), ) or with the input itself Our goal is to identify each part: identify how the output is related to itself and to the input, ε 5 / 21

112 Time Series Sequence of observations taken sequentially over time If the variable has randomness, the sequence is stochastic process w realization 1 realization 2 realization Time (k) w(k) = w(k 1) + ε(k) / 21

113 Description of Stochastic Processes Two Relevant Questions: 1 Does probability of an outcome (or realization) of w(k + τ) depend on outcome of w(k)? Are w(k) and w(k + τ) are independent? 2 Does the distribution of w(k) or the joint distribution of {w(k), w(k + τ)} depend on k? Does the mean change with time? Are the covariances, cov{w(1), w(5)} and cov{w(11), w(15)}, different? In the last lecture, we learned Autocovariance Weakly Stationary Process / 21

114 In the Context of Our Applications We assume ``weakly stationary processes" constant means constant variances autocovariances depend only on lags Autocovariance R w (τ) = E{(w(k) w)(w(k + τ) w) T } 8 / 21

115 Autocorrelation Autocovariance has scale Normalized quantity: autocorrelation ρ w (τ) = cov{w(k), w(k + τ)} var(w(k + τ)) var(w(k)) = E{(w(k) w)(w(k + τ) w)t } σ 2 w = R w(τ) R w (0) Note: because of the stationarity, var(k + τ) = var(k) = σ 2 w 9 / 21

116 Autocorrelation & Autocovariance 1 Variance is simply the autocovariance at lag 0 σ 2 w = R w (0) 2 Autocorrelation and autocovariance are symmetric in lag τ R w (τ) = R w ( τ) ρ w (τ) = ρ w ( τ) 3 Autocorrelation is bounded and normalized 1 ρ w (τ) 1 4 Autocorrelation and autocovariance are parameters summarizing the probability behaviour of the stochastic process w(k) Sample autocorrelation / autocovariance using sample data 10 / 21

117 Disturbance Example 1 w(k) = ε(k) + 05ε(k 1) where ε N (0, σ 2 ε ) Autocorrelations? Lag 0: ρ w (0) = 1 Current output always perfectly correlated with itself Lag 1: ρ w (1) = 048 Lag > 1: ρ w (τ > 1) = 0 1 ρw(τ) τ / 21

118 Disturbance Example 1 Non-zero values to lag 1 Lag 1 moving average disturbance = MA(1) disturbance 3 Disturbance Ex Notice local "trends" w Time (k) Time Response 12 / 21

119 Disturbance Example 2 Dependence on past output Autocorrelations lag 0: ρ w (0) = 1 lag 1: ρ w (1) = 0 lag 2: ρ w (2) = (0) 2 = 03 lag k: ρ w (k) = (0) k w(k + 1) = 0w(k) + ε(k + 1) 1 ρw(τ) τ 13 / 21

120 Disturbance Example 2: Autoregressive The disturbance w is weakly stationary Sum of stationary stochastic processes; an infinite sum of the white noise sequence ε(k)'s AR coefficient is 0 convergent Mean is zero and variance is constant 3 Disturbance Ex Notice local trends w Time (k) 14 / 21

121 Two examples were Example 1 is a moving average disturbance: w(k) = ε(k) + 05ε(k 1) = {1 θ 1 q 1 }ε(k) Example 2 is an autoregressive disturbance: w(k) = 0w(k 1) + ε(k) = 1 ε(k) 1 ϕ 1 q 1 15 / 21

122 Detecting Model Structure from Data Given time series data of a stochastic process Examine autocorrelation plot If a sharp cut-off at lag k is detected, then the disturbance is a moving average, order k, disturbance If a gradual decline is observed, then the disturbance contains an autoregressive component Long tails indicate either a higher-order autoregressive component, or a pole near 1 If the autocorrelations alternate in positive and negative values one or more of the roots is negative 1 / 21

123 Estimating Autocovariances from Data Sample autocovariance function ˆR w (τ) = 1 N τ (w(k) w)(w(k + τ) w) T N k=1 N is the number of data points ˆR w (0) is sample variance of w(k) When ˆR w (τ) is computed, confidence limits should be considered Sample autocorrelation function ˆρ w (τ) = ˆR w (τ) ˆR w (0) Confidence limits for the autocorrelation are derived by examining how variability propagates through calculations 1 / 21

124 Example 1: Estimated Autocorrelation Plot Sharp cut-off: Moving Average 12 Moving Average Disturbance Process 1 08 autocorrelation lag 18 / 21

125 Example 2: Estimated Autocorrelation Plot Gradual decay: Autoregressive 12 Autoregressive Disturbance Process 1 08 autocorrelation lag 19 / 21

126 Partial Autocorrelation It is difficult to identify the order of the AR component due to the gradual decay for the AR structure Q How do we find the order of the AR component? A Partial Autocorrelation: Compute autocorrelation between w(k) and w(k + τ) after taking into account the dependence on values k + 1, k + 2, k + τ 1 The partial autocorrelation at lag τ is the autocorrelation between w(k) and w(k + τ) that is not accounted for by lags 1 through τ 1 The partial autocorrelation of an AR(τ) process is zero at lag τ + 1 and greater If the sample autocorrelation plot indicates that an AR model is appropriate, the sample partial autocorrelation is examined to identify the order of an AR model 20 / 21

127 How to Use the Partial Autocorrelation For an autoregressive process of order p, a sharp cut-off will be observed after lag p in which the partial autocorrelations go to zero No more explicit dependence beyond lag p The partial autoregressive plot for moving average processes will exhibit a decay The autocorrelation and partial autocorrelation behaviours are dual for autoregressive and moving average processes 21 / 21

128 SysID Example Jong Min Lee

129 Autocorrelation Function (ACF) Explains the dependency of samples Correlation between x(k) and x(k+τ) What can we infer? Impulse-type ACF: white noise Sharp cut-off after n lags: MA(n) Tails off to several lags: AR Very slow decrease: Integrated process eg, η(k+1) = η(k) + e(k+1) 2

130 Partial Autocorrelation Function (PACF) Reveals what cannot be explained by ACF Useful for AR(n) processes What we can infer: Impulse-type PACF: white noise signal Sharp cut-off after n lags: AR(n) Tails off to several lags: MA 3

131 Identification of Linear I/O Models H(q) e(k) or η(k) u(k) G(q) y p (k) + + y d (k) y(k) Given u(k) and y(k), determine a plant (deterministic) model (G) and a disturbance (noise, stochastic) model (H) 4

132 5 Box-Jenkins Structure ) ( ) ( ) ( ) ( ) ( ) ( ) ( k e D q q C nk k u q F q B k y + = ) ( = nb b nb q q b b q B! nf f nf q q f q F =! ) ( doc bj C = 1, D = F: ARX C = 1, D = 1: OE

133 Output Error (OE) Models B( q) y ( k) = u( k nk) + e( k) F( q) Yields best unbiased estimate of the plant model Disturbance model is not considered Useful for identifying plant models in BJ oe(iddata, [nb nf nk])

134 OE Models Assumes that the disturbance (or the prediction errors) is white The best OE model is what restricts the correlations between residuals and input within a confidence bound ACF(residuals) need not be white because the assumption of OE may not be true If ACF(res) is white, then the system posesses OE structure If ACF(res) is non-white, build an ARMA model to the residuals to obtain a disturbance model

135 General Procedure for Identification ARX ARMAX OE + ARMA BJ 8

136 Identification Example

137 Example 4 y1 0 Impulse response estimate Time Lab 2 example: IO data time lags 4000 samples Unknown process order, time-delay, and structure Detrending: detrend(data, constant ) Divide data set into training and validation sets cra: impulse response coefficients and time-delay (nk = 2) 0 10

138 Step Response From u1 >> z = iddata(y, u); >> step(z) To y y starts increasing from t=2 G( s) 1e 35s s Time Discrete-time pole: exp(-(1/35)x1) = 0 No imaginary pole: discrete-time poles are real and positive Step response is used to compare with that of the identified model 11

139 Fitting an ARX Model A ( q) y( k) = B( q) u( k nk) + e( k) model=arx(data, [na nb nk]) Fitting plant + disturbance models simultaneously Use identified time-delay for nk nk = pure delay + zoh delay (1) Start with low orders Relevant functions: arxstruc, selstruc 12

140 ARX Models resid(marx112, ztd) resid(marx442, ztd) 1 Correlation function of residuals Output y1 1 Correlation function of residuals Output y lag Cross corr function between input u1 and residuals from output y lag lag Cross corr function between input u1 and residuals from output y lag 13

141 ARX 442 Measured Output and Simulated Model Output Measured Output marx442 Fit: 898% 18 1 From u y1-1 To y Time 0 Process ARX 442 Model Time 14

142 OE for Plant Model (G p ) Correlation function of residuals Output y lag Cross corr function between input u1 and residuals from output y b1 resid(oe112, ztd) y ( k) = u( k 2) + e( k) 1 1+ f q lag 1 OK We can model the residuals later with disturbance modelling (AR, MA ) Not Acceptable! u and y are still correlated further room to improve 15

143 OE Model: [2 2 2] If correlation is not satisfactory, increase numerator or denominator order [nb nf nk]: [1 1 2] [1 2 2] [2 1 2] [2 2 2] 1 Correlation function of residuals Output y lag Cross corr function between input u1 and residuals from output y1 01 ACF of residuals is almost white (Not a concern here However, there may not be a need to construct a disturbance model) lag Acceptable! 1

144 OE222 Model >> present(moe222) Discrete-time IDPOLY model: y(t) = [B(q)/F(q)]u(t) + e(t) B(q) = ( ) q^ ( ) q^-3 F(q) = (+-0012) q^ ( ) q^-2 Gain: 015 (G(1)) Poles: and 0392 (>>roots([ ])) 1

145 Comments Use training data for checking residuals (resid) Use validation data for checking prediction performance (compare) Compare model s response on the validation data set: compare(ztd, moe222) step(ztd, moe222, r*- ); legend( process, OEmodel ) g=spa(ztd); bode(g, moe222, r- ) 18

146 Measured Output and Simulated Model Output 4 3 Measured Output moe222 Fit: 892% y From u1 to y Time 18 From u1 Amplitude Phase (degrees) Frequency (rad/s) Process OE222 To y Process moe Time 19

147 OK We finished modelling the plant (G) using OE Are we really done?

148 Disturbance Modelling: G d? y d = y y p (residuals from the OE222 model) 1 Plot ACF and PACF of the residuals (y d ) 2 Determine whether the disturbance process (G) is MA, AR, or ARMA 3 Use the MATLAB function ARMAX doc armax If data has no input channels and just one output channel (that is, it is a time series), then orders = [na nc] A ( q) y( k) = C( q) e( k) and armax calculates an ARMA model for the time series 21

149 Identifying Gd 12 Auto-correlation Function 12 Partial-autocorrelation Function lag lag MA(1) or AR(1)?: AR is a more general I will choose AR(1) [na nc] = [1 0] >> erroe = pe(ztd, moe222); >> autocf(erroe, 20, 0); pautocf(erroe, 20); >> mdist = armax(erroey, [1 0]); >> present(mdist) Discrete-time IDPOLY model: A(q)y(t) = e(t) A(q) = ( ) q^-1 22

150 The residuals of y d from G d errmdist = pe(mdist, erroe); figure; [R3, CI3, NR3] = autocf(errmdisty, 20, 0); 12 Auto-correlation Function lag 23

151 Putting Together: BJ Model You can refine G and H by putting them together in BJ (see the manual) In this case, OE and ARMA modelling steps provide an initial estimate for the BJ structure One can directly use the BJ by specifying model orders 24

152 45804 Process Dynamics & Control Lecture : Linear Quadratic Control Deterministic Case Jong Min Lee Chemical & Biomolecular Engineering Seoul National University May 19, / 22

153 Outline Basic problem setup Deterministic system Stochastic system 2 / 22

154 Basic Problem Setup Linear Deterministic System: x(k + 1) = Ax(k) + Bu(k) (1) y(k) = Cx(k) (2) We consider time-invariant system for simplicity For a linear state feedback controller The closed-loop response is: u(k) = L(k)x(k) (3) x(k + 1) = (A BL(k))x(k) Stability The state feedback controller (3) stabilizes the system if all the eigenvalues of (A BL) lie within the unit disk 3 / 22

155 Objective of LQ A system visits a sequence of states of x(0), x(1),, x(p), and desired sequence of states x(0), x(1),, x(p) Without loss of generality, the desired trajectory, x, can be set as the origin Objective function p 1 [ min x T (k)qx(k) + u T (k)ru(k) ] + x T (p)q t x(p) (4) k=0 Q and R are symmetric positive definite; Q t is positive semi-definite Q provide relative importance to the errors in various states R accounts for the cost of implementing input moves if p =, it is infinite horizon problem 4 / 22

156 Open-Loop Control vs Feedback Control Optimal open-loop control problem Find the optimal sequence of u(0),, u(k) for given (as a function of) distribution of x(0) Optimal feedback control problem Find the optimal feedback law u(k) = f(x(k)) or u(k) = f(y(k), y(k 1), ) For completely deterministic systems, the two should provide the same performance State Feedback vs Output Feedback u(k) = f(x(k)) State feedback u(k) = F(y(k)) Output feedback F would be a dynamic operator in general 5 / 22