Part 1: Introduction to the Algebraic Approach to ILC

Transcription

1 IEEE ICMA 2006 Tutorial Workshop: Control Algebraic Analysis and Optimal Design Presenters: Contributor: Kevin L. Moore Colorado School of Mines YangQuan Chen Utah State University Hyo-Sung Ahn ETRI, Korea IEEE 2006 International Conference on Mechatronics and Automation LuoYang, China 25 June

2 Outline Introduction Control System Design: Motivation for ILC Control: The Basic Idea Some Comments on the History of ILC ILC Problem Formulation The Supervector Notation The w-transform: z-operator Along the Repetition Axis ILC as a MIMO Control System Repetition-Domain Poles Repetition-Domain Internal Model Principle The Complete Framework Repetition-Varying Inputs and Disturbances Model Variation Along the Repetition Axis 2

3 Control Design Problem Reference Error Input System to be Output controlled Given: Find: Such that: System to be controlled. (using feedback). 1) Closed-loop system is stable. 2) Steady-state error is acceptable. 3) Transient response is acceptable. 3

4 Motivation for the Problem of Control Transient response design is hard: 2.5 1) Robustness is always an issue: - Modelling uncertainty. - Parameter variations. - Disturbances. 2) Lack of theory (design uncertainty): - Relation between pole/zero locations and transient response. - Relation between Q/R weighting matrices in optimal control and transient response. - Nonlinear systems. Many systems of interest in applications are operated in a repetitive fashion. Control (ILC) is a methodology that tries to address the problem of transient response performance for systems that operate repetitively

5 Systems that Execute the Same Trajectory Repetitively Step 1: Robot at rest, waiting for workpiece. Step 2: Workpiece moved into position. Step 3: Robot moves to desired location Step 4: Robot returns to rest and waits for next workpiece. 5

6 Errors are Repeated When Trajectories are Repeated A typical joint angle trajectory for the example might look like this: Each time the system is operated it will see the same overshoot, rise time, settling time, and steady-state error. learning control attempts to improve the transient response by adjusting the input to the plant during future system operation based on the errors observed during past operation. 6

7 Control Standard iterative learning control scheme: u k System y k Memory Memory Memory u k+1 y d A typical ILC algorithm has the form: u k+1 (t) = u k (t) + γe k (t + 1). Standard ILC assumptions include: Stable dynamics or some kind of Lipschitz condition. System returns to the same initial conditions at the start of each trial. Each trial has the same length. 7

8 Trial (k-1) Trial k Trial (k+1) Error t-1 t t+1 t-1 t t+1 t-1 t t+1 Input t-1 t t+1 t-1 t t+1 t-1 t t+1 (a) ILC: u 1 ( t) = u ( t) + f ( e ( + k k 1)) k + t Error t-1 t t+1 t-1 t t+1 t-1 t t+1 Input t-1 t t+1 t-1 t t+1 t-1 t t+1 (b) Conventional feedback: u k 1 ( t ) = f ( e + 1 ( + k t 1 )) 8

9 Example 1 Consider the plant: We wish to force the system to follow a signal y d : y(t + 1) =.7y(t).012y(t 1) + u(t) y(0) = 2 y(1) = 2 9

10 Example 1 (cont.) Use the following ILC procedure: 1. Let 2. Run the system 3. Compute 4. Let 5. Iterate u 0 (t) = y d (t) e 0 (t) = y d (t) y 0 (t) u 1 (t) = u 0 (t) + 0.5e 0 (t + 1) Each iteration shows an improvement in tracking performance (plot shows desired and actual output on first, 5th, and 10th trials and input on 10th trial). 10

11 Example 1 (cont.) 11

12 Example 2 For the nominal plant: x k+1 = [ ] x 1 0 k + [ ] 0.5 u 1 k y k = [1, 0.5]x k Track the reference trajectory: Y d (j) = sin(8.0j/100) We use the standard Arimoto algorithm: u k+1 (t) = u k (t) + γe k (t + 1) with four different gains: γ = 0.5, γ = 0.85, γ = 1.15, γ =

13 Example 2 (cont.) γ = 0.5 γ = 0.85 Y d and Y k Y d and Y k Discrete time point γ = Discrete time point γ = 1.5 Y d and Y k Y d and Y k Discrete time point Discrete time point For each gain, the ILC algorithm converges, but the convergence rate depends on γ. Without knowing an accurate model of the plant, we achieve perfect tracking by iteratively updating the input from trial to trail. 13

14 Example 3 Consider a simple two-link manipulator modelled by: where A(x k )ẍ k + B(x k, ẋ k )ẋ k + C(x k ) = u k x(t) = (θ 1 (t), θ 2 (t)) ( T ) cos θ cos θ A(x) = cos θ ( ).135 sin θ2 0 B(x, ẋ) =.27 sin θ 2.135(sin θ 2 ) ( θ 2 ) sin θ sin(θ C(x) = 1 + θ 2 ) sin(θ 1 + θ 2 ) u k (t) = vector of torques applied to the joints g m 1 θ 2 l 1 l 2 m 2 θ 1 l 1 = l 2 = 0.3m m 1 = 3.0kg m 2 = 1.5kg 14

15 Example 3 (cont.) Define the vectors: y k y d = (x T k, ẋ T k, ẍ T k ) T = (x T d, ẋ T d, ẍ T d ) T The learning controller is defined by: u k = r k α k Γy k + C(x d (0)) r k+1 α k+1 = r k + α k Γe k = α k + γ e k m Γ is a fixed feedback gain matrix that has been made time-varying through the multiplication by the gain α k. r k can be described as a time-varying reference input. r k (t) and adaptation of α k are effectively the ILC part of the algorithm. With this algorithm we have combined conventional feedback with iterative learning control. 15

16 Example 3 (cont.) 16

17 ILC History ILC surveys: K. L. Moore, M. Dahleh, and S. P. Bhattacharyya. learning control: a survey and new results. J. of Robotic Systems, 9(5): , K. L. Moore. learning control - an expository overview. Applied & Computational Controls, Signal Processing, and Circuits, 1(1): , H. S. Ahn, Y. Q. Chen, and K. L. Moore. learning control: brief survey and categorization IEEE Trans. on Systems, Man, and Cybernetics, Accepted to appear. recent CSM paper by Allyene Pioneering work: United States Patent 3,555,252 Control of Actuators in Control Systems, filed 1967, awarded 1971, learned characteristics of actuators and used this knowledge to correct command signals. J. B. Edwards. Stability problems in the control of linear multipass processes. Proc. IEE, 121(11): , M. Uchiyama. Formulation of high-speed motion pattern of a mechanical arm by trial. Trans. SICE (Soc. Instrum. Contr. Eng.), 14(6): (in Japanese), S. Arimoto, S. Kawamura, and F. Miyazaki. Bettering operation of robots by learning. J. of Robotic Systems, 1(2): ,

18 Edwards, Proc. IEE (1974) 18

19 First ILC paper- in Japanese (1978) 19

20 First ILC paper- in English (1984) 20

21 ILC Research History ILC has a well-established research history: More than 1000 papers: 90 Number of publication Journal papers Conference papers Year At least four monographs. Over 20 Ph.D dissertations. 21

22 Robots Robots ILC Research History (cont.) Rotary systems Rotary systems Process control Miscellaneous Process control Bio-applications Semiconductors Miscellaneous Bio-applications Actuators Power systems Semiconductors Actuators Power systems Miscellaneous Typical By application areas Structure Update rule Robust Optimal Mechanical nonlinearity Adaptive Neural ILC for control By theoretical areas. 22

23 Selected ILC Industrial Applications ILC patents in hard disk drive servo: YangQuan Chen s US6,437,936 Repeatable runout compensation using a learning algorithm with scheduled parameters. YangQuan Chen s US6,563,663 Repeatable runout compensation using iterative learning control in a disc storage system. Robotics: Michael Norrlöf s patent on ABB robots. US Path correction for an industrial robot. Gantry motion control: Work by Southampton Sheffield Control (SSILC) Group. 23

24 Control Engineering - History and ILC Prehistory of automatic control Primitive period Classical control Modern control Classic control Nonlinear control Estimation Robust control Optimal control Adaptive control Intelligent control H_inf Interval Fuzzy Neural Net ILC... 24

25 ILC Problem Formulation Standard iterative learning control scheme: u k System y k Memory Memory Memory u k+1 y d Goal: Find a learning control algorithm so that for all t [0, t f ] u k+1 (t) = f L (previous information) lim y k(t) = y d (t) k We will consider this problem primarily for discrete-time, linear systems. 25

26 Some Control Algorithms Arimoto first proposed a learning control algorithm of the form: Convergence is assured if I CBΓ i < 1. u k+1 (t) = u k (t) + Γė k (t) Arimoto has also considered more general algorithms of the form: u k+1 = u k + Φe k + Γė k + Ψ e k dt Various researchers have used gradient methods to optimize the gain G k in: u k+1 (t) = u k (t) + G k e k (t + 1) It is also useful for design to specify the learning control algorithm in the frequency domain, for example: U k+1 (s) = L(s)[U k (s) + ae k (s)] Many schemes in the literature can be classified with one of the algorithms given above. 26

27 LTI ILC Convergence Conditions Theorem: For the plant y k = T s u k, the linear time-invariant learning control algorithm converges to a fixed point u (t) given by with a final error defined on the interval (t 0, t f ) if Observation: u k+1 = T u u k + T e (y d y k ) u (t) = (I T u + T e T s ) 1 T e y d (t) e (t) = lim k (y k y d ) = (I T s (I T u + T e T s ) 1 T e )y d (t) If T u = I then e (t) = 0 for all t [t o, t f ]. Otherwise the error will be non-zero. T u T e T s i < 1 27

28 LTI Control - Nature of the Solution Question: Given T s, how do we pick T u and T e to make the final error e (t) as small as possible, for the general linear ILC algorithm: u k+1 (t) = T u u k (t) + T e (y d (t) y k (t)) Answer: Let T n solve the problem: min T n (I T s T n )y d It turns out that we can specify T u and T e in terms of T n and the resulting learning controller converges to an optimal system input given by: u (t) = T ny d (t) Conclusion:The essential effect of a properly designed learning controller is to produce the output of the best possible inverse of the system in the direction of y d. 28

29 LTI ILC - Solution Details OPT1: Let u k U, y d, y k Y and T s, T u, T e X. Then given y d and T s, find Tu and Te solve min (I T s(i T u + T e T s ) 1 T e )y d T u,t e X subject to T u T e T s i < 1. that OPT2: Let y d Y and let T n, T s X. Then given y d and T s, find T n that solve min (I T st n )y d T n X Theorem: Let Tn be the solution of OPT2. Factor Tn = TmT e where Tm 1 Define Tu = I Tm 1 + Te T s. Then Tu and Te are the solution of OPT1. X and I T 1 m i < 1. If we plug these into the expression for the fixed-point of the input to the system we find: u (t) = T ny d (t) Note: The factorization in the Theorem can always be done, with the result that u (t) = T ny d (t). 29

30 Outline Introduction Control System Design: Motivation for ILC Control: The Basic Idea Some Comments on the History of ILC ILC Problem Formulation The Supervector Notation The w-transform: z-operator Along the Repetition Axis ILC as a MIMO Control System Repetition-Domain Poles Repetition-Domain Internal Model Principle The Complete Framework Repetition-Varying Inputs and Disturbances Model Variation Along the Repetition Axis 30

31 ILC as a Two-Dimensional Process Suppose the plant is a scalar discrete-time dynamical system, described as: where y k (t + 1) = f S [y k (t), u k (t), t] k denotes a trial (or execution, repetition, pass, etc.). t [0, N] denotes time (integer-valued). y k (0) = y d (0) = y 0 for all k. Use a general form of a typical ILC algorithm for a system with relative degree one: where e k (t) = y d (t) y k (t) is the error on trial k. y d (t) is a desired output signal. u k+1 (t) = f L [u k (t), e k (t + 1), k] 31

32 ILC as a Two-Dimensional Process (cont.) Combine the plant equation with the ILC update rule to get: y k+1 (t + 1) = f S [y k (t), u k+1 (t), t] = f S [f L [y k (t), u k (t), e k (t + 1), k], t] Changing the notation slightly we get: Clearly this is a 2-D system: y(k + 1, t + 1) = f[y k (t), u(k, t), e(k, t + 1), k, t] Dynamic equation indexed by two variables: k and t. k defines the repetition domain (Longman/Phan terminology). t is the normal time-domain variable. But, ILC differs from a complete 2-D system design problem: One of the dimensions (time) is a finite, fixed interval, thus convergence in that direction (traditional stability) is always assured for linear systems. In the ILC problem we admit non-causal processing in one dimension (time) but not in the other (repetition). We can exploit these points to turn the 2-D problem into a 1-D problem. 32

33 The Supervector Framework of ILC Consider an SISO, LTI discrete-time plant with relative degree m: Y (z) = H(z)U(z) = (h m z m + h m+1 z (m+1) + h m+2 z (m+2) + )U(z) By lifting along the time axis, for each trial k define: U k Y k Y d = [u k (0), u k (1),, u k (N 1)] T = [y k (m), y k (m + 1),, y k (m + N 1)] T = [y d (m), y d (m + 2),, y d (m + N 1)] T Thus the linear plant can be described by Y k = H p U k where: h h 2 h H p = h 3 h 2 h h N h N 1 h N 2... h 1 The lower triangular matrix H p is formed using the system s Markov parameters. Notice the non-causal shift ahead in forming the vectors U k and Y k. 33

34 The Supervector Framework of ILC (cont.) For the linear, time-varying case, suppose we have the plant given by: x k (t + 1) = A(t)x k (t) + B(t)u k (t) y k (t) = C(t)x k (t) + D(t)u k (t) Then the same notation again results in Y k = H p U k, where now: H p = h m, h m+1,0 h m, h m+2,0 h m+1,1 h m, h m+n 1,0 h m+n 2,1 h m+n 3,2... h m,n 1 The lifting operation over a finite interval allows us to: Represent our dynamical system in R 1 into a static system in R N. 34

35 The Update Law Using Supervector Notation Suppose we have a simple Arimoto-style ILC update equation with a constant gain γ: In our R 1 representation, we write: u k+1 (t) = u k (t) + γe k (t + 1) In our R N representation, we write: U k+1 = U k + ΓE k where Γ = γ γ γ γ 35

36 The Update Law Using Supervector Notation (cont.) Suppose we filter with an LTI filter during the ILC update: In our R 1 representation we would have the form: u k+1 (t) = u k (t) + L(z)e k (t + 1) In our R N representation we would have the form: U k+1 = U k + LE k where L is a Topelitz matrix of the Markov parameters of L(z), given, in the case of a causal, LTI update law, by: L = L m L m+1 L m L m+2 L m+1 L m L m+n 1 L m+n 2 L m+n 3... L m 36

37 The Update Law Using Supervector Notation (cont.) We may similarly consider time-varying and noncausal filters in the ILC update law: U k+1 = U k + LE k A causal (in time), time-varying filter in the ILC update law might look like, for example: n 1, n 2,0 n 1, L = n 3,0 n 2,1 n 1, n N,0 n N 1,1 n N 2,2... n 1,N 1 A non-causal (in time), time-invariant averaging filter in the ILC update law might look like, for example: K K K K K K L = K K K K K 37

38 The Update Law Using Supervector Notation (cont.) The supervector notation can also be applied to other ILC update schemes. For example: The Q-filter often introduced for stability (along the iteration domain) has the R 1 representation: u k+1 (t) = Q(z)(u k (t) + L(z)e k (t + 1)) The equivalent R N representation is: U k+1 = Q(U k + LE k ) where Q is a Toeplitz matrix formed using the Markov parameters of the filter Q(z). 38

39 The ILC Design Problem The design of an ILC controller can be thought of as the selection of the matrix L: For a causal ILC updating law, L will be in lower-triangular Toeplitz form. For a noncausal ILC updating law, L will be in upper-triangular Toeplitz form. For the popular zero-phase learning filter, L will be in a symmetrical band diagonal form. L can also be fully populated. Motivated by these comments, we will refer to the causal and non-causal elements of a general matrix Γ as follows: Γ = γ 11 γ 12 γ γ 1N γ 21 γ 22 γ 23 noncausal γ 2N γ 31 γ 32 γ γ 3N. causal..... γ N1 γ N2 γ N3... γ NN The diagonal elements of Γ are referred to as Arimoto gains. 39

40 w-transform: the z-operator in the Iteration Domain Introduce a new shift variable, w, with the property that, for each fixed integer t: w 1 u k (t) = u k 1 (t) For a scalar x k (t), combining the lifting operation to get the supervector X k with the shift operation gives what we call the w-transform of x k (t), which we denote by X(w) Then the ILC update algorithm: u k+1 (t) = u k (t) + L(z)e k (t + 1) which, using our supervector notation, can be written as U k+1 = U k + LE k can also be written as: wu(w) = U(w) + LE(w) where U(w) and E(w) are the w-transforms of U k and E k, respectively. Note that we can also write this as where. E(w) = C(w)U(w) C(w) = 1 (w 1) L 40

41 ILC as a MIMO Control System The term C(w) = 1 (w 1) L is effectively the controller of the system (in the repetition domain). This can be depicted as: 41

42 Higher-Order ILC in the Iteration Domain We can use these ideas to develop more general expressions ILC algorithms. For example, a higher-order ILC algorithm could have the form: which corresponds to: u k+1 (t) = k 1 u k (t) + k 2 u k 1 (t) + γe k (t + 1) C(w) = γw w 2 k 1 w k 2 Next we show how to extend these notions to develop an algebraic (matrix fraction) description of the ILC problem. 42

43 A Matrix Fraction Formulation Suppose we consider a more general ILC update equation given by (for relative degree m = 1): u k+1 (t) = D n (z)u k (t) + D n 1 (z)u k 1 (t) + + D 1 (z)u k n+1 (t) + D 0 (z)u k n (t) +N n (z)e k (t N n 1 (z)e k 1 (t N 1 (z)e k n+1 (t + 1) + N 0 (z)e k n (t + 1) which has the supervector expression U k+1 = D n U k + D n 1 U k D 1 U k n+1 + D 0 U k n +N n E k + N n 1 E k N 1 E k n+1 + N 0 E k n Aside: note that there are a couple of variations on the theme that people sometimes consider: U k+1 = U k + LE k+1 U k+1 = U k + L 1 E k + L 0 E k+1 These can be accomodated by adding a term N n+1 E k+1 in the expression above, resulting in the so-called current iteration feedback, or CITE. 43

44 A Matrix Fraction Formulation (cont.) Applying the shift variable w we get: D c (w)u(w) = N c (w)e(w) where D c (w) = Iw n+1 D n 1 w n D 1 w D 0 N c (w) = N n w n + N n 1 w n N 1 w + N 0 This can be written in a matrix fraction as U(w) = C(w)E(w) where: C(w) = 1 D c (w)n c (w) Thus, through the addition of higher-order terms in the update algorithm, the ILC problem has been converted from a static multivariable representation to a dynamic (in the repetition domain) multivariable representation. Note that we will always get a linear, time-invariant system like this, even if the actual plant is time-varying. Also, because D c (w) is of degree n + 1 and N c (w) is of degree n, we have relative degree one in the repetition-domain, unless some of the gain matrices are set to zero. 44

45 ILC Convergence via Repetition-Domain Poles From the figure we see that in the repetition-domain the closed-loop dynamics are defined by: G cl (w) = H p [I + C(w)H p ] 1 C(w) = H p [ D c (w) + N c (w)h p ] 1 N c (w) Thus the ILC algorithm will converge (i.e., E k a constant) if G cl is stable. Determining the stability of this feedback system may not be trivial: It is a multivariable feedback system of dimension N, where N could be very large. But, the problem is simplified due to the fact that the plant H p is a constant, lower-triangular matrix. 45

46 Repetition-Domain Internal Model Principle Because Y d is a constant and our plant is type zero (e.g., H p is a constant matrix), the internal model principle applied in the repetition domain requires that C(w) should have an integrator effect to cause E k 0. Thus, we modify the ILC update algorithm as: U k+1 = (I D n 1 )U k + (D n 1 D n 2 )U k 1 + +(D 2 D 1 )U k n+2 + (D 1 D 0 )U k n+1 + D 0 U k n +N n E k + N n 1 E k N 1 E k n+1 + N 0 E k n Taking the w-transform of the ILC update equation, combining terms, and simplifying gives: where (w 1)D c (w)u(w) = N c (w)e(w) D c (w) = w n + D n 1 w n D 1 w + D 0 N c (w) = N n w n + N n 1 w n N 1 w + N 0 46

47 Repetition-Domain Internal Model Principle (cont.) This can also be written in a matrix fraction as: but where we now have: U(w) = C(w)E(w) C(w) = (w 1) 1 D 1 c (w)n c (w) For this update law the repetition-domain closed-loop dynamics become: ( G cl (w) = H I + I (w 1) C(w)H ) 1 I (w 1) C(w), = H[(w 1)D c (w) + N c (w)h] 1 N c (w) Thus, we now have an integrator in the feedback loop (a discrete integrator, in the repetition domain) and, applying the final value theorem to G cl, we get E k 0 as long as the ILC algorithm converges (i.e., as long as G cl is stable). 47

48 Outline Introduction Control System Design: Motivation for ILC Control: The Basic Idea Some Comments on the History of ILC ILC Problem Formulation The Supervector Notation The w-transform: z-operator Along the Repetition Axis ILC as a MIMO Control System Repetition-Domain Poles Repetition-Domain Internal Model Principle The Complete Framework Repetition-Varying Inputs and Disturbances Model Variation Along the Repetition Axis 48

49 Higher-Order ILC in the Iteration Domain, Revisited A key feature of our matrix fraction, algebraic framework is that it assumes use of higher-order ILC. At the 02 IFAC World Congress a special session explored the value that could be obtained from such algorithms: One possible benefit could be due to more freedom in placing the poles (in the w-plane). It has been suggested in the literature that such schemes can give faster convergence. However, we can show dead-beat convergence using any order ILC. Thus, higher-order ILC can be no faster than first-order. One conclusion from the 02 IFAC special sessions is that higher-order ILC is primarily beneficial when there is repetition-domain uncertainty. Several such possibilities arise: Iteration-to-iteration reference variation. Iteration-to-iteration disturbances and noise. model variation from repetition-to-repetition. The matrix fraction, or algebraic, approach can help in these cases. 49

50 Iteration-Varying Disturbances In ILC, it is assumed that desired trajectory y d (t) and external disturbance are invariant with respect to iterations. When these assumptions are not valid, conventional integral-type, first-order ILC will no longer work well. In such a case, ILC schemes that are higher-order along the iteration direction will help. Consider a stable plant H a (z) = z 0.8 (z 0.55)(z 0.75) Let the plant be subject to an additive output disturbance d(k, t) = 0.01( 1) k 1 This is an iteration-varying, alternating disturbance. If the iteration number k is odd, the disturbance is a positive constant in iteration k while when k is even, the disturbance jumps to a negative constant. In the simulation, we wish to track a ramp up and down on a finite interval. 50

51 Example: First-Order ILC output signals desired output output at iter. #60 2 norm of the input signal time (sec.) Iteration number The Root Mean Square Error Iteration number h(t) N h 1 =1 and sum hj j=2 = t u k+1 (t) = u k (t) + γe k (t + 1), γ = 0.9 C(w) = 1 (w 1) L 51

52 Example: Second-Order, Internal Model ILC 1 5 output signals desired output output at iter. #60 2 norm of the input signal time (sec.) Iteration number The Root Mean Square Error Iteration number h(t) N h 1 =1 and sum hj j=2 = t u k+1 (t) = u k 1 (t) + γe k 1 (t + 1) with γ = 0.9 C(w) = 1 (w 2 1) L 52

53 Example 2 - Iteration-Domain Ramp Disturbance Consider a stable plant H a (z) = Assume that y d (t) does not vary w.r.t. iterations. z 0.8 (z 0.55)(z 0.75). However, we add a disturbance d(k, t) at the output y k (t). In iteration k, the disturbance is a constant w.r.t. time but its value is proportional to k. Thus In the simulation, we set c 0 = d(k, t) = c 0 k 53

54 Example 2: First-Order ILC output signals desired output output at iter. #60 2 norm of the input signal time (sec.) Iteration number The Root Mean Square Error Iteration number h(t) N h =1 and sum hj = j= t u k+1 (t) = u k (t) + γe k (t + 1), γ =

55 Example 2: Second-Order, Internal Model ILC output signals desired output output at iter. #60 2 norm of the input signal time (sec.) Iteration number The Root Mean Square Error Iteration number h(t) N h =1 and sum hj = j= t u k+1 (t) = 2u k (t) u k 1 (t) + γ(2e k (t + 1) e k 1 (t + 1)), γ =

56 A Complete Design Framework We have presented several important facts about ILC: The supervector notation lets us write the ILC system as a matrix fraction, introducing an algebraic framework. In this framework we are able to discuss convergence in terms of pole in the iteration-domain. In this framework we can consider rejection of iteration-dependent disturbances and noise as well as the tracking of iteration-dependent reference signals (by virtue of the internal model principle). In the same line of thought, we can next introduce the idea of iteration-varying models. 56

57 Iteration-Varying s Can view the classic multi-pass (Owens and Edwards) and linear repetitive systems (Owens and Rogers) as a generalization of the static MIMO system Y k = H p U k into a dynamic (in iteration) MIMO system, so that Y k+1 = A 0 Y k + B 0 U k becomes H(w) = (wi A 0 ) 1 B 0 Introduce iteration-varying plant uncertainty, so the static MIMO system Y k = H p U k becomes a dynamic (in iteration) and uncertain MIMO system, such as or or or H p = H 0 (I + H) H p [H, H] H p = H 0 + H(w) H p (w) = H 0 (w)(i + H(w)) etc. 57

58 Complete Framework D (w) N(w) Y d (w) - E(w) C ( w ILC ) U (w) 1 ( w 1) H p (w) Y (w) C CITE (w) ΔH(w) Y d (w), D(w) and N(w) describe, respectively, the iteration-varying reference, disturbance, and noise signals. H p (w) denotes the (possibly iteration varying) plant. H p (w) represents the uncertainty in plant model, which may also be iteration-dependent. C ILC (w) denotes the ILC update law. C CITE (w) denotes any current iteration feedback that might be employed. The term 1 (w 1) denotes the natural one-iteration delay inherent in ILC. 58

59 Categorization of Problems Y d D N C H Y d (z) 0 0 Γ(w 1) 1 H p Classical Arimoto algorithm Y d (z) 0 0 Γ(w 1) 1 H p (w) Owens multipass problem Y d (z) D(w) 0 C(w) H p General (higher-order) problem Y d (w) D(w) 0 C(w) H p General (higher-order) problem Y d (w) D(w) N(w) C(w) H p (w) + (w) Most general problem Y d (z) D(z) 0 C(w) H(w) + (w) Frequency-domain uncertainty Y d (z) 0 w(t), v(t) Γ(w 1) 1 H p Least quadratic ILC Y d (z) 0 w(t), v(t) C(w) H p Stochastic ILC (general) Y d (z) 0 0 Γ(w 1) 1 H I Interval ILC Y d (z) D(z) w(t), v(t) C(w) H(z) + H(z) Time-domain H problem Y d (z) D(w) w(k, t), v(k, t) C(w) H(w) + H(w) Iteration-domain H ILC Y d (z) 0 w(k, t), v(k, t) Γ(k) H p + H(k) Iteration-varying uncertainty and control Y d (z) 0 H Γ Hp Intermittent measurement problem

60 Outline Introduction Control System Design: Motivation for ILC Control: The Basic Idea Some Comments on the History of ILC The Supervector Notation The w-transform: z-operator Along the Repetition Axis ILC as a MIMO Control System Repetition-Domain Poles Repetition-Domain Internal Model Principle The Complete Framework Repetition-Varying Inputs and Disturbances Model Variation Along the Repetition Axis 60