Two-dimensional (2D) Systems based Iterative Learning Control Design

Transcription

1 Two-dimensional (2D) Systems based Iterative Learning Control Design Electronics and Computer Science University of Southampton Southampton, SO17 1BJ, UK

2 Contents Stability and Performance 2D Systems and Repetitive Processes ILC Design Frequency Domain Inequalities based Design

3 Stability and Performance Design for trial-to-trial error convergence is sometimes not enough also need to control along the trial performance. Consider G(s) = (s + 1)(s + 5) (s + 3)(s 2 + 4s + 29) which is to be controlled in the ILC setting using the P-type law where L = 3 gives trial to trial error convergence (1) u k+1 (t) = u k (t) + Le k+1 (t) (2) (a) Input Progression (b) Error Progression (c) Output Progression Time (s) 2 0 Trial Number Time (s) 2 0 Trial Number Time (s) 2 0 Trial Number

4 Stability and Performance 100 (a) Input (Trial: 30) 1 (a) Error (Trial: 30) 1.4 (a) Output (Trial: 30) Time (s) Time (s) Reference Output Time (s) If along the trial performance is an issue, one way to proceed is to design a feedback control law for the plant and then apply lifting based ILC to the controlled system. This is a two-stage design process! An alternative is to use 2D systems theory, which is the subject of this section.

5 2D Systems and Repetitive Processes

6 2D Systems Introduction Many processes make repeated executions of the same task or operation over a finite interval. One possible scenario: the operation is completed and then the process is rest to the initial location ready for the start of the next execution. Call each operation a pass and the output produced the pass profile. Such systems were initially termed multipass processes, subsequently changed to repetitive processes, to describe the operation of longwall coal cutting where the novel feature is that the pass profile produced on the previous pass acts as a forcing function on, and hence contributes to, the next pass profile.

7 Coal Cutting as an Example In longwall coal cutting, the coal cutting machine is hauled along the entire length of the coal face riding on the semi-flexible structure of the armoured face conveyor, denoted A. F. C., which transports away the coal cut by the rotating drum. In the simplest mode of operation, these machines only cut in one direction, and they are hauled back in reverse at high speed for the start of each new pass of the coal face. Between passes, the conveyor is snaked forward using hydraulic rams such that the machine now rests on the newly cut floor, i.e., the pass profile produced during the previous pass.

8 Coal Cutting as an Example Side Elevation NUCLEONIC COAL SENSOR MACHINE BODY ALONG FACE DIRECTION COAL CUTTING DRUM STEERING JACK A.F.C. STONE Figure 1: Side-elevation of a longwall coal cutting machine.

9 Coal Cutting as an Example Plan View NEW COAL FACE SENSOR DRUM OLD COAL FACE MACHINE BODY A. F. C Figure 2: Plan view of a longwall coal cutting machine.

10 Coal Cutting as an Example Pushover Phase NEW COAL FACE LOOSELY JOINED A.F.C. PANS PUSHING RAMS Figure 3: Snaking of the conveyor during the pushover stage.

11 1 Stone/coal interface 2 Cut roof 3 Stone 4 Coal seam 16 9 J k+ 1(t) 5 Drum 15 β k+ 1(t) 6 Floor sensor 14 skid D 7 Cut floor 13 Y k+1(t) 8 Interface 17 Z k+ 1(t) 18 e k+ 1(t + R) 21 e k+ 1(t + R + F) 11 t X 12 t 19 t + R 20 t + R + F Figure 4: Side elevation with variables labeled.

12 Simplified Model of the Coal Cutting Example Suppose that all angular deflections are small. Then elementary geometrical considerations immediately yield the following description of the coal cutting process dynamics over 0 t α Y k+1 (t) + Z k+1 (t) =e k+1 (t + R) + W γ k+1 (t + R) + Rβ k+1 (t) + J k+1 (t) (3) where γ, β denote the transverse and longitudinal tilts of the machine, respectively, and Z k+1 (t) denotes the height of the stone/coal interface above the same fixed datum plane as the A.F.C.

13 Simplified Model of the Coal Cutting Example The transverse and longitudinal tilts of the machine are also those of the supporting conveyor structure and are given by and γ k+1 (t) = (e k+1(t) e k (t)) W β k+1 (t) = (e k+1(t) e k+1 (t + F )) (5) F respectively. Suppose also that the A. F. C. moulds itself exactly onto the cut floor on which it rests. Then (k 2 is a positive real constant) e k+1 (t) = k 2 (Y k (t) + Z k (t)) (6) (4)

14 Control of Coal Cutting A possible control law is J k+1 (t) = k 1 (R k+1 (t) Y k+1 (t X)) W γ k+1 (t), 0 t α (7) where k 1 is a positive real constant and R k+1 (t) is a new external reference vector taken to represent the desired coal thickness on pass k + 1. Suppose that the variable Z k+1 (t) is set equal to zero. Then the closed-loop description of the process dynamics over 0 t α, k 0, is Y k+1 (t) = k 1 Y k+1 (t X) + k 2 Y k (t) + k 1 R k+1 (t), X > 0 (8) with assumed pass initial conditions Y k+1 (t) = 0, X t 0 (9)

15 Closed-loop Response k 1 = 0.8, k 2 = 1, X = 1.25, α = 10, R k+1 (t) = 1. k t/x -1.0 Figure 5: Closed-loop system negative unit step response.

16 Unique Control Problem The oscillations grow, or increase in amplitude, severely from pass to pass, i.e., in the k direction. Consequently the deterioration in system performance after the first pass must be due to the effects of the cut floor profile on the previous pass, i.e., the output dynamics, on any pass acts, by the basic system geometry, as a forcing function, or disturbance, on, and hence contributes to, the dynamics of the next pass, i.e., the shape of the floor profile produced on the next pass of the cutting machine along the coal face.

17 Unique Control Problem This interaction between successive pass profile dynamics is the essential unique characteristic of all repetitive processes and in cases such as that of Figure 5 appropriate control action is required. The question now is: how should control laws be designed to prevent these oscillations from arising? First we briefly consider an approach developed from a practical standpoint for the coal cutting example.

18 Classical Control Approach (SISO) If the example under consideration is single-input single-output (SISO) and the dynamics are assumed to be linear, an obvious intuitive approach to stability analysis and controller design is to make use of well known classical tools such as the inverse Nyquist criterion. The essence of such an approach is to use the single variable V := kα + t to convert the process under consideration into an equivalent infinite length single pass process where the relationships between variables are expressed only in terms of V, termed the total distance traversed. In particular, a variable, say, Y k+1 (t), k 0, is identified as a function Y (V ) of V defined for 0 V <.

19 Classical Control Approach (SISO) Applying this approach to (8) (9) yields Y (V ) = k 1 Y (V X) + k 2 Y (V α) + k 1 R(V ) (10) and this repetitive process is said to be stable if and only if the system of (10) is stable in the standard, or 1D, linear systems sense. Accepting this premise, the original repetitive process dynamics are now amenable to analysis by 1D technique. Hence, for example, taking the Laplace transform wrt V and applying the inverse Nyquist criterion, (10) is stable if and only if k 1 < 1 k 2 (11)

20 Classical Control Approach (SISO)? In order to apply classical (frequency domain or otherwise) 1D linear systems theory and controller design techniques to linear repetitive processes it is necessary to make the following assumptions. The pass length α is long (but finite) and hence the effects of the initial conditions at the start of each pass can be ignored. The effects of the previous pass dynamics can be represented by a long delay term, e.g., k 2 Y (V α) in (10).

21 Classical Control Approach (SISO)? It is easily shown, using a discretized version of (3)-(9) with Y k+1 (t X), X t 0, appropriately chosen, that the initial conditions on each pass have a crucial effect on the performance of the (simplified) long-wall coal cutting dynamics. This, in turn, suggests that for processes with a lag (X) on the current pass, analysis based on the concept of the total distance traversed is valid only in the range kα + X V (k + 1)α, k 0 and only in the following range for processes with no lag in the current pass dynamics kα V (k + 1)α, k 0

22 Classical Control Approach Conclusions The classically based approach to stability analysis and controller design for linear repetitive processes, as outlined above for the long-wall coal cutting example, is critically limited by the following major factors. It completely ignores the effects of the initial conditions on each pass the structure of these conditions have a critical effect on process stability. It does not form the basis for the development of a rigorous generally applicable stability analysis with onward development into the specification and design of control schemes. Next some more physical examples are noted.

23 Soil Compaction

24 Web forming Delivery mechanism CD Actuator array Cross direction (CD) Material web Machine direction (MD) Scanning gauge Gauge path

25 Metal rolling yk-1 ( t ) y k ( t)

26 Welding

27 Digging using a Shovel

28 Repetitive Processes The above examples are physically based. Repetitive process theory can also be used to facilitate control-related analysis in other areas. One example is iterative solution algorithms for nonlinear dynamic optimal control problems using the maximum principle. Another is the design of ILC laws that provide both trial-to-trial error convergence, and along the trial stability. E. Rogers, K. Galkowski and D. H. Owens Control Systems Theory and Applications for Linear Repetitive Processes Series: Lecture Notes in Control and Information Sciences. Springer Verlag, Vol. 349, 2007.

29 A General Model Ingredients Explicit retention of the effects of the initial conditions on each pass. Inclusion of all (if possible) linear constant pass length processes as special cases. Clearly any general model of repetitive processes must, as an essential requirement, include all the unique features of these processes. Considering first the most general case, i.e., nonlinear dynamics and a variable pass length, the unique features of a repetitive process can be summarized as follows.

30 A General Model Ingredients A number of passes, indexed as k 0, through a set of dynamics. Each pass is characterized by a pass length α k, and a pass profile y k (t) defined on 0 t α k, where y k (t) can be a vector or scalar quantity. An initial pass profile y 0 (t) defined on 0 t α 0, where α 0 is the initial pass length. The function y 0 (t) together with the initial conditions on each pass form the initial, or boundary, conditions for the process. Each pass will be subject to its own disturbances and control inputs.

31 A General Model Ingredients The process is unit memory, i.e., the dynamics on pass k + 1 (explicitly) depend only on the independent inputs to that pass and the pass profile on the previous pass k. Assumption: Linear dynamics and a constant pass length, i.e., α k = α, k 0. Note: Examples do exist, for example, metal rolling where a constant pass length is an assumption. Relatively little work has yet been reported on nonlinear repetitive processes.

32 General Model Formal Statement Definition A linear repetitive process of constant pass length α > 0 consists of a Banach space E α, a linear subspace W α of E α, and a bounded linear operator L α mapping E α into itself (also written L α B(E α, E α )). The system dynamics are described by linear recursion relations of the form y k+1 = L α y k + b k+1, k 0 (12) where y k E α is the pass profile on pass k and b k+1 W α. The term L α y k in this model represents the contribution from pass k to pass k + 1 and b k+1 represents initial conditions, disturbances and control input effects on this pass.

33 Differential Linear Repetitive Processes ẋ k+1 (t) =Ax k+1 (t) + Bu k+1 (t) + B 0 y k (t) y k+1 (t) =Cx k+1 (t) + Du k+1 (t) + D 0 y k (t) (13) where on pass k, x k (t) is the n 1 state vector, y k (t) is the m 1 pass profile vector, and u k (t) is the l 1 vector of control inputs. Boundary conditions: The simplest possible form for these is x k+1 (0) =d k+1, k 0 y 0 (t) =f(t), 0 t α (14)

34 Differential Linear Repetitive Processes Chose E α as the space of bounded continuous mappings of the interval 0 t α into R m with norm max 0 t α y(t) m where m is any convenient norm on R m. Then (L α y)(t) = C t and b k+1 over 0 t α, k 0, by b k+1 = Ce At d k+1 + C 0 e A(t τ) B 0 y(τ) dτ + D 0 y(t), 0 t α (15) t 0 e A(t τ) Bu k+1 (τ) dτ + Du k+1 (t) (16) If d k+1 lies in a subspace W of R n, W α E α is obtained by evaluating (16) for all d k+1 and all admissible u k+1.

35 Discrete Linear Repetitive Processes x k+1 (p + 1) =Ax k+1 (p) + Bu k+1 (p) + B 0 y k (p) y k+1 (p) =Cx k+1 (p) + Du k+1 (p) + D 0 y k (p) (17) where on pass k, x k (p) is the n 1 state vector, y k (p) is the m 1 vector pass profile, and u k (p) is the l 1 vector of control inputs. Boundary Conditions: The simplest possible form for these is x k+1 (0) =d k+1, k 0 y 0 (p) =f(p), 0 p α (18)

36 Discrete Linear Repetitive Processes In this case, set E α = l m 2 [0, α] the space of all real m 1 vectors of length α (corresponding to p = 1, 2,, α). Then over 1 p α, p 1 (L α y)(p) = CA p 1 r B 0 y(r) + D 0 y(p) r=0 and over 1 p α, k 0, p 1 b k+1 = CA p d k+1 + CA p 1 r Bu k+1 (r) + Du k+1 (p) r=0

37 Other Examples and Boundary Conditions Other physical examples that can be written as special cases of the abstract model are given in Rogers, Galkowski and Owens (2007). These include metal/material rolling and classes of delay differential systems. Next we examine the boundary conditions in more detail as it turns out these have a critical influence on stability and hence control. We only consider the discrete case the differential case can be found in the above reference.

38 Boundary Conditions The state initial vector on each pass can be a function of points along the previous pass hence known as dynamic boundary conditions. One form x k+1 (0) = d k+1 + N K j y(p j ) (19) j=1 where 0 p 1 < p 2 < < p N α are N sample points along the previous pass and K j, 1 j N, is an n m matrix with constant entries. It is a simple exercise to show that this model is also a special case of the abstract model in E α = l m 2 [0, α].

39 Boundary Conditions It is known that an example that is stable under the x k+1 (0) of (18) can be unstable under x k+1 (0) of (32). Hence under-modeling x k+1 (0) can have disastrous consequences. The classical approach of bolting passes together using the total distance traversed variable can also lead to difficulties. Dynamic boundary conditions also widen the class of systems that can be written as repetitive processes, e.g., classes of delay systems, and iterative solution algorithms for dynamic nonlinear optimal control problems based on the maximum principle.

40 Discrete Processes and 2D Discrete Linear Systems Discrete linear repetitive processes share certain structural similarities with well known and extensively studied 2D quarter plane causal discrete linear systems modeled by the Roesser (1975) and Fornasini-Marchesini (1978) state-space models (in all their various forms). Also there are obvious structural links between this class of linear repetitive processes and 1D discrete linear time-invariant systems. Do there exist 2D Roesser/Fornasini-Marchesini and/or 1D discrete linear time invariant systems equivalent state-space model descriptions of the dynamics of discrete linear repetitive processes?

41 2D Roesser Model x h (i + 1, j) =A 1 x h (i, j) + A 2 x v (i, j) + B 1 u(i, j) x v (i, j + 1) =A 3 x h (i, j) + A 4 x v (i, j) + B 2 u(i, j) (20) In this model i and j are the positive integer valued horizontal and vertical coefficients, x h is the n 1 vector of horizontally transmitted information, x v is the m 1 vector of vertically transmitted information, and u is the l 1 vector of control inputs. The static output equation is omitted here.

42 2D Fornasini Marchesini Model In the Fornasini-Marchesini model, the state vector is not split into horizontal and vertical components. Again, the output equation is not required in this work and, with z(i, j) denoting the appropriately dimensioned state vector, the basic model of this type has the structure z(i + 1, j + 1) =A 5 z(i + 1, j) + A 6 z(i, j + 1) + A 7 z(i, j) + B 3 u(i + 1, j) + B 4 u(i, j + 1) (21) where, as in (20), u is the (appropriately dimensioned) control input vector.

43 Discrete Processes/2D Discrete Linear Systems The essential difference between discrete linear repetitive processes and 2D linear systems described by the Roesser (and hence Fornasini-Marchesini) state-space models is that in the former case information propagation in one of the two separate directions, along the pass, only occurs over the fixed finite duration of the pass length. Hence, despite initial structural similarities, (20) is not an equivalent 2D linear systems Roesser state-space model description of the dynamics of discrete linear repetitive processes. See Rogers, Galkowski and Owens (2007), and the relevant cited references in this research monograph, for a full treatment of this issue.

44 Visualization of updating structures N N ' W /53 a Figure 6: Illustrating the updating structures of 2D discrete linear systems and discrete linear repetitive processes (LRPs).

45 Updating structures again k k+2 k+1 x k+1 (p) x k+1 (p+1) k y k (0) y k (p) y k (p+1) y k (α 1) k 0 α 1 p k+2 k+1 x k+1 (p) y k+1 (p) k y k (0) y k (p) y k (p+1) y k (α 1) 0 α 1 p Figure 7: Illustrating the updating structures of discrete linear repetitive processes.

46 Updating structures again Initial profile Points on pass ( p) y (p) k 0 Pass initial conditions Passes ( k) Figure 8: Illustrating the updating structure of discrete linear repetitive processes.

47 Abstract Model based Stability Theory Unique control problem is oscillations that increase in amplitude in the pass-to-pass direction. This strongly suggests a Bounded-Input Bounded-Output (BIBO) based stability theory. Intuitively: a bounded initial pass profile y 0 should produce bounded sequences of pass profiles {y k } k. Must not forget the finite pass length so we can have BIBO stability over the finite pass length or uniformly with respect to the pass length. Based on the abstract model introduced previously in these notes, i.e., y k+1 = L α y k + b k+1, k 0, (22) where y k E α is the pass profile on pass k and b k+1 W α.

48 Asymptotic (or pass-to-pass) Stability Definition The linear repetitive process (22) is said to be asymptotically stable over the finite pass length α if a real scalar δ > 0 such that, given any initial profile y 0 and any disturbance sequence {b k } k 1 bounded in norm, i.e., b k c 1 for some c 1 0 and k 1, the output sequence generated by the perturbed process y k+1 = (L α + γ)y k + b k+1, k 0 (23) is bounded in norm whenever γ δ.

49 Asymptotic (or pass-to-pass) Stability is used to denote the norm on E α and also the induced operator norm. The idea behind the the L α + γ structure is motivated by the physical idea that stability should be retained if the model is perturbed slightly to (try to) capture modeling errors etc. Route map from here: use (relatively elementary) tools from the theory of Banach spaces to characterize stability and then interpret for particular cases of interest. Complete theoretical development in Rogers, Galkowski and Owens (2007).

50 Asymptotic (or pass-to-pass) Stability Theorem The linear repetitive process (22) is asymptotically stable if and only if r(l α ) < 1 (24) where r( ) denotes the spectral radius. In ILC, we are interested (at least mathematically) in what happens as k (trial-to-trial error convergence). Definition Suppose that the linear repetitive process (22) is asymptotically stable and let {b k } k 1 be a disturbance sequence that converges strongly to a disturbance b. Then the corresponding limit profile is y := lim k y k

51 Asymptotic (or pass-to-pass) Stability Theorem Suppose that the linear repetitive process (22) is asymptotically stable and let {b k } k 1 be a disturbance sequence that converges strongly to a disturbance b. Then the limit profile corresponding to this disturbance sequence is the unique solution of the linear equation y = L α y + b (25) or where I denotes the identity operator in E α. y = (I L α ) 1 b (26)

52 Asymptotic (or pass-to-pass) Stability Application To apply this theory requires the computation of the spectral radius of the bounded linear operator L α. In general, this is a non-trivial task. For the cases of interest here r(l α ) is obtaining by first computing the eigenvalues of a matrix, and then evaluating their moduli to obtain for a q q matrix, say, r i, i = 1, 2,..., q. r(l α ) = max 1 i q r i (27)

53 Asymptotic Stability Application Theorem Discrete linear repetitive processes described by (17) and (18) are asymptotically stable if, and only if, r(d 0 ) < 1 (28) Limit Profile x (p + 1) = (A + B 0 (I D 0 ) 1 C)x (p) +(B + B 0 (I D 0 ) 1 D)u (p) y (p) = (I D) 1 Cx (p) + (I D 0 ) 1 Du (p) x (0) = d (29)

54 Asymptotic Stability An Example Consider the case when A = 0.5, B = 1, B 0 = β, C = 1, D = D 0 = 0 where β is a real scalar. This example is asymptotically stable with limit profile y (p + 1) = βy (p) + u (p) Hence if β 1 then the limit profile is unstable. How to fix this problem?

55 Stability Along the Pass In the above example r(a) < 1. The conditions r(d 0 ) < 1 and r(a) < 1 correspond to practical stability (first developed for 2D discrete linear systems). Practical stability is not enough in general! The alternative is stability along the pass, i.e., the BIBO property uniformly with respect to the pass length. In abstract model terms stability along the pass requires the existence of finite real numbers M > 0 and λ (0, 1), independent of α, such that L k α M λ k, for all k

56 Stability Along the Pass Theorem The linear repetitive process (22) is stable along the pass if, and only if, (a) (b) and M 0 := sup α>0 for some real number λ (r, 1). r := sup r(l α ) < 1 (30) α>0 sup (zi L α ) 1 < (31) z λ

57 Stability Along the Pass Theorem Suppose that the pair {A, B 0 } is controllable and the pair {C, A} observable. Then discrete linear repetitive processes described by (17) and (18) are stable along the pass if and only if (a) r(d 0 ) < 1, r(a) < 1 (b) all eigenvalues of the transfer-function matrix G(z 1 ) = C(z 1 I A) 1 B 0 + D 0 have modulus strictly less than unity for all z 1 = 1.

58 Stability Along the Pass The third condition here describes the coupling between the previous and current pass profiles. With zero state initial vector sequence and zero input y k (z 1 ) = G k (z 1 )y 0 (z 1 ) Hence this condition requires that the complete frequency content of the initial pass profile is attenuated from pass-to-pass. All conditions in the above theorem can be tested by direct application of standard, as termed 1D in the multidimensional systems literature, linear systems tests.

59 Dynamic Boundary Conditions Theorem Discrete linear repetitive processes described by (17) and x k+1 (0) = d k+1 + N K j y(j) (32) are asymptotically stable if and only if the roots of α 1 det zi K j z(zi D 0 ) 1 C(A + B 0 (zi D 0 ) 1 C) j = 0 j=0 have modulus strictly less than unity. j=1 (33)

60 Comparison with Classical Approach Key Point: The boundary conditions alone can destroy stability asymptotic and hence along the pass. The classical approach does not pick this up. The coal cutting model (8) with boundary conditions including (9) is asymptotically stable when k 2 (0, 1). This conclusion is not valid unless (9) is true.

61 Stability Along the Pass Lyapunov Function/LMI Interpretation A Lyapunov function interpretation of stability along the pass leads to LMI based tests for stability along the pass. These are sufficient but not necessary but do lead to control law design algorithms. They extend to robust control and also to the design of ILC control laws. Notation: A symmetric positive definite matrix, say H, is denoted by H > 0. (A symmetric negative definite matrix is denoted H < 0.)

62 Stability Along the Pass Lyapunov Function/LMI Interpretation This approach covers both differential and discrete processes. Here we consider discrete processes described by (17) and (18). Candidate Lyapunov function V (k, p) = V 1 (k, p) + V 2 (k, p) = x T k+1 (p)p 1x k+1 (p) + y T k (p)p 2y k (p) (34) The matrices P 1 > 0 and P 2 > 0 are of dimensions n n and m m respectively.

63 Stability along the pass Lyapunov function/lmi interpretation Define the increment for the Lyapunov function as where V (k, p) = V 1 (k, p) + V 2 (k, p) V 1 (k, p) = x k+1 (p + 1)P 1 x k+1 (p + 1) x k+1 (p)p 1 x k+1 (p) V 2 (k, p) = y T k+1 (k, p)p 2y k+1 (p) y T k (p)p 2y k (p) Theorem A discrete linear repetitive process described by (17) and (18) is stable along the pass if the Lyapunov function (34) satisfies V (k, p) < 0 (35)

64 Stability along the pass LMI interpretation Theorem Discrete linear repetitive processes described by (17) and (18) are stable along the pass if there exists a matrix P = diag {P 1, P 2 } > 0 such that the following LMI is feasible: [ P M T ] P < 0 (36) P [ ] A B0 where M = C D 0 Note: Many equivalent versions of this result exist.

65 ILC Design

66 ILC design Objectives To obtain an algorithm that guarantees trial-to-trial error convergence, and along the trial stability. the approach is as follows: formulate the problem in a linear repetitive process setting and employ the existing stability theory for these processes, and develop the stability analysis to allow LMI based control law design. To be consistent with the ILC literature, the word pass is now replaced by the word trial.

67 Repetitive process ILC design Work in the discrete domain under ZOH sampling at a uniform rate T s seconds to produce a discrete state-space model with matrices {A, B, C} Consider a control law of the form u k+1 (p) = u k (p) + u k+1 (p) (37) where u k+1 (p) is the change in the control vector from one trial to the next. Also introduce, for analysis purposes only η k+1 (p + 1) = x k+1 (p) x k (p) (38)

68 Repetitive process ILC design Control law and u k+1 (p) = K 1 µ k+1 (p + 1) + K 2 µ k+1 (p) + K 3 e k (p + 1), µ k (p) = y k (p 1) y k 1 (p 1) = Cη k (p). The term K 1 µ k+1 (p + 1) is added added to compensate for the effects of not assuming that the state vector is available for use in the control law only the output is used.

69 Repetitive process ILC design With η k+1 (p + 1) = [ ηk+1 (p + 1) η k+1 (p) the controlled system dynamics can be written as η k+1 (p + 1) = Â η k+1(p) + ˆB 0 e k (p) e k+1 (p) = Ĉ η k+1(p) + ˆD 0 e k (p) (39) where the matrices are given on the next page. This is a discrete linear repetitive process state-space model of the form (17). Hence repetitive process stability theory can be applied to this ILC control scheme. ]

70 Repetitive process ILC design [ ] A + BK1 C BK Â = 2 C [ I ] 0 BK3 ˆB 0 = 0 Ĉ = [ CA CBK 1 C CBK 2 C ] ˆD 0 = [I CBK 3 ]

71 Asymptotic stability vs. stability along the trial Asymptotic stability (Rogers, Galkowski and Owens 2007) BIBO stability over the finite trial length or duration α > 0. This property holds if and only if r( ˆD 0 ) < 1, i.e., r(i CBK 3 ) < 1. This last condition is precisely that obtained by applying 2D Roesser model based discrete linear systems stability theory (Kurek and Zaremba 1993) to (39). Still no along the trial control. Still a problem when the first Markov parameter (CB) is zero! Example Switch to stability along the trial, i.e., BIBO stability uniformly, i.e., independent of the trial duration.

72 Stability along the trial Theorem A discrete linear repetitive process described by (39) is stable along the pass if there exist matrices Y > 0 and Z > 0 such that the following LMI holds Y Z ( ) ( ) 0 Z ( )  1 Y  2 Y Y < 0 where  1 = [  ˆB0 0 0 ] [ 0 0, Â2 = Ĉ ˆD 0 ]

73 Stability along the trial Ω 1 = Ω 2 = AY 1 + BN 1 C BN 2 C BN 3 Y CAY 1 CBN 1 C CBN 2 C Y 3 CBN 3

74 Stability along the trial Theorem A discrete linear repetitive process described by (39) is along the pass if there exist matrices Y > 0 and Z > 0 such that the following LMI holds Z Y ( ) ( ) 0 Z ( ) < 0 Ω 1 Ω 2 Y CY 1 = P C, CY 2 = QC where Y = diag {Y 1, Y 2, Y 3}. If the LMI with equality constraints is feasible, the control law matrices can be computed as K 1 = N 1P 1 K 2 = N 2Q 1 K 3 = N 3Y 1 3

75 Final Control Law Structure The control law can be written as u k (p) = u k 1 (p) + K 1 (y k (p) y k 1 (p)) + K 2 (y k (p 1) y k 1 (p 1)) + K 3 (y r (p + 1) y k 1 (p + 1)) (40) The last term on the right-hand side is ILC phase-lead. The second and third terms are proportional in nature, where the third is ILC phase-lead. A form of HOILC. Setting K 1 = 0 and K 2 = 0 recovers the control law in Kurek and Zaremba (1993).

76 Experimental Results Gantry Robot A Kurek and Zaremba design K 1 = 0, K 2 = 0 and K 3 = 100. Totally unacceptable as the simulations show cannot be implemented. Figure 9: The input, error and output progression.

77 Experimental Results Gantry Robot Output (m) Along the trial performance (Trial number: 5) Reference Output Input (V) Error (m) Time (s) Figure 10: The output on trial 4 (red line) compared to y r (blue) together with the input (middle plot) and the error (bottom).

78 Initial tests Control law parameters (after tuning) selected as: K 1 = , K 2 = , K 3 = Mean Squared Error for X axis Iteration Number Figure 11: Error begins to build up on later trials.

79 Initial tests 2 Input for trial Output for trial x 10 4 Error for trial Iteration Number Figure 12: The input/output/error plots for the 60th trial in one experiment.

80 Initial tests 2.5 x 10 7 Error Frequency Spectrum 2 Amplitude Filter Frequency Response 0 Magnitude (db) Frequency (Hz) Figure 13:The error frequency spectrum.

81 Zero-phase filtering In ILC once a trial is completed all of the data is available and hence zero-phase filtering can be completed in the time taken to reset to the starting location for the start of the next trial. Implementation: First put the data through a filter as usual. This attenuates frequencies above the cut-off and adds phase shift (lag). Now time reverse the data and put it through the filter again. This doubles the attenuation and removes the phase shift.

82 Filtering To remove the high-frequency buildup a Chebyshev 6th order zero phase filter was applied. The cutoff frequency was hand-tuned in each case to give the best performance: Example Filtering too strong cutoff frequency too low longer learning (more trials) worse performance; larger lowest possible error Filtering too weak cutoff frequency too high. divergence of the error unacceptable.

83 Filtering 10 3 Mean Squared Error for X axis Cut off frequency: 5 Hz Cut off frequency: 15 Hz Iteration Number Figure 14: A comparison of different filters.

84 Results with good filter design Figure 15: Output sequence after applying a filter with cut off frequency 10 Hz.

85 Results with good filter design Figure 16: Input sequence after applying a filter with cut off frequency 10 Hz.

86 Results with good filter design Figure 17: Error sequence after applying a filter with cut off frequency 10 Hz.

87 Performance along the trial 2 Input for trial Input (V) Output (m) Output Reference Output for trial x 10 4 Error for trial Error (m) Sample Number Figure 18: Tracking performance along the 100th trial.

88 Comparison of tuning K 3 To compare the effects of selecting different control law matrices, consider (in all cases T s = 0.05) 1. K 3 = (K 1 = , K 2 = ) 2. K 3 = (K 1 = , K 2 = ) 3. K 3 = (K 1 = , K 2 = ) Mean Squared Error for X axis K 3 = K 3 = K 3 = Iteration Number Figure 19: Tracking performance along the 100th trial.

89 Extensions This setting extends to allow design with extensively used uncertainty descriptions, such as additive uncertainty in the plant state-space model matrices. This permits the effects of uncertainty in the along the trial dynamics. The lifting approach results in matrix product terms arising from the uncertainty description (not easy/impossible to handle in this setting). Zero Markov Parameters: As with all other design methods the results above do not apply if the first Markov parameter CB = 0. The analysis above can be extended to this case but at the expense of losing control at the first instance along the trial. If the first non-zero Markov parameter is h > 1 then no control is possible over the first h instances along the trial. Also the case with lifting designs.

90 Zero Markov Parameters If the first non-zero Markov parameter is h > 1 then no control is possible over the first h instances along the trial. Also the case with lifting designs. Next, recent results using the Kalman-Yakubovich-Popov (KYP) lemma are introduced that deal directly with the case of a zero Markov parameter the first for ease of presentation.

91 Frequency Domain Inequalities based Design

92 Introduction to the KYP lemma Frequency domain inequalities (FDI) have played a crucial role in describing design specifications for feedback control synthesis based, for example, on Bode or Nyquist plots. Due to the infinite frequency range, however, FDIs are not directly useful for rigorous analysis and design of control systems. Question How can FDIs can be formulated mathematically and is it possible to convert them into numerically tractable procedures?

93 Introduction to the KYP lemma For differential linear time-invariant systems, an FDI an be formulated as G(jω) ΠG(jω) < 0, ω R where Π is a real symmetric matrix and G(s) = C(sI A) 1 B + D is a matrix valued, real-rational transfer-function matrix. Problem Infinite frequency range.

94 KYP lemma The KYP lemma states that, given the matrices {A, B, C, D} and a Hermitian matrix (equal to its complex conjugate transpose) Θ, the FDI [ (jωi A) 1 ] [ B (jωi A) Θ 1 ] B < 0 I I holds for all ω R { } if and only if the following LMI holds [ A B I 0 ] [ P 0 0 P ] [ A B I 0 ] + Θ 0 A. Rantzer On the Kalman-Yakubovich-Popov lemma Systems and Contorl Letters, 28(1):7 10, 1996.

95 KYP lemma, cont d Main features The infinitely many inequalities parameterized by ω can be checked by solving a finite-dimensional convex feasibility problem. Appropriate choices of Θ allows tghe characterization of various system properties, including positive-realness and bounded-realness. This standard KYP lemma deals with FDIs over the entire frequency spectrum and it is not completely compatible with practical design specifications given over a finite frequency range.

96 Examples of FDI specifications Digital filter design Consider the problem of finding a stable SISO transfer function F (z) satisfying a set of frequency domain specifications. Typical design requirements for band-pass filters are given in the following Chebyshev approximation setting F (e jθ ) ɛ l : θ ϑ l F (e jθ ) ɛ h : ϑ h θ π F (e jθ ) M(e jθ ) ɛ p : ϑ 1 θ ϑ 2 F (e jθ ) 1 + ɛ 0 : θ where ϑ l < ϑ 1 < ϑ 2 < ϑ h and M(z) is a given function that has desired gain/phase properties in the pass-band [ϑ 1, ϑ 2 ]. An ideal response M(z) would have the unity magnitude and the linear phase in the pass-band, i.e., M(z) = z d.

97 Examples of FDI specifications Open-loop shaping Given an SISO plant P (s), a typical set of specifications on a controller K(s) is given in terms of the Nyquist plot of the open-loop transfer function as L(s) := K(s)P (s) high-gain in the low frequency range for sensitivity reduction and reference tracking adequate stability margins and bandwidth maximization in the middle-frequency range, and small gain, i.e., roll-off, in the high-frequency range for robust stability. See the next figure.

98 Examples of FDI specifications Open-loop shaping, cont d

99 Existing solutions There are two main approaches to solving control problems over a finite frequency range 1. A low/band/high-pass filter is added to the system in series as a weighting that emphasizes a particular frequency range and then the design parameters are chosen such that the weighted system norm is small. The deficiencies are: system complexity and, in particular, the controller order is increased, and the process of selecting appropriate weights is tedious and can be time-consuming. 2. Frequency axis griding - FDIs are approximated by a finite number of FDIs at selected frequency points but it is difficult to: determine a priori how fine the grid should be to achieve a certain performance. impose performance guarantees in the design process as the violation of the specifications may occur at a frequency between the grid points.

100 Towards a generalized KYP lemma Frequency set characterization The frequency set is characterized by a quadratic equation and inequality of the form { [ ] [ ] [ ] [ ] } λ λ λ λ Λ(Φ, Ψ) := λ C : Φ = 0, Ψ By an appropriate choice of Φ and Ψ, the set Λ can be specialized to define a certain range for the frequency variable λ. For example, in the continuous-time setting and low frequency range λ Λ LF, i.e., Λ LF := {jω ω R, ω ϖ} [ ] [ Φ =, Ψ = ϖ 2 ]

101 The Generalized KYP lemma Let Π, Φ, Ψ be given and let G(λ) be a rational function G(λ) = C(λI A) 1 B + D then the parameterized inequality condition G(λ) ΠG(λ) < 0, for all λ Λ(Φ, Ψ) holds if and only if there exist matrices P and Q > 0 such that [ A B I 0 ] [ A B (Φ P + Ψ Q) I 0 ] + Θ < 0

102 The Generalized KYP lemma denotes the matrix Kronecher product. T. Iwasaki and S. Hara Generalized KYP lemma: unified frequency domain inequalities with design applications IEEE Transactions on Automatic Control, 50(1):41 59, 2005.

103 Finite frequency specification It can be shown that G(jω) < γ, for all ω l ω ω h is equivalent to [ G(jω) 1 and by the GKYP lemma [ A B I 0 Π ] {}}{ T [ ][ ] 1 0 G(jω) 0 γ 2 < 0, for all ω 1 l ω ω h [{}}{ Q P +jω c Q P +jω c Q ω l ω h Q ] T P ΨP +QΦQ where ω c =(ω l +ω h )/2. ] [ A B I 0 ] + [ ] T C D Π 0 1 [ ] C D 0 0 1

104 Towards design Procedures via the KYP Lemma Define F T A = [ A T I C T ], F T B = [ B T 0 D T ] then there exists W such that [I F B ] Ω [I F B ] < F A W R + (F A W R) where [ ] f(q, P, Ξ) 0 Ω = 0 Π Let A, B, C and D be the matrices of a state-space realization of a linear continuous-time system. let Π andξ be given matrices which describe convex regions and (semi) finite frequency ranges Q > 0, P,W, R are matrix variables to be found

105 GKYP ILC design L + + feedforward loop e k(p) Q memory r(p) + - C + + P y k(p) feedback loop

106 GKYP ILC design Let the input on trial k + 1 to the plant P in the previous figure be u k+1 = u c k+1 + f k+1 where u c k+1 is the output of C and Hence e k = f k+1 = Q(f k + Le k ) P C r P 1 + P C f k and the error propagation with r = 0 is e k+1 = S P f k+1 = Q(1 S P L)e k

107 GKYP ILC design where f k = Sp 1 e k is used under the assumption that S 1 P exists and S P denotes the sensitivity function S P = P 1 + P C This ILC algorithm is stable (and error convergence occurs) if Q(z)(1 L(z)S P (z)) < 1, z = 1

108 GKYP ILC design The key task is to select the filters Q and L. The most obvious choice is L = S 1 P Q-filter can improve the learning process and robustness (design of such a filter whose magnitude is unity for the frequency range where reference tracking is required and zero at frequencies outside this range) Note also that such a Q-filter has zero-phase and therefore allows for high-frequency attenuation without introducing any lag. In practice, such a filter frequency response cannot be physically realized but it can be closely approximated.

109 GKYP ILC design The basic inequality for synthesis is (when Q = 1) ) σ max (G(e jθ ) = G(e jθ ) < 1, θ Θ where G = 1 LS P. Also it is routine to show that where γ = 1. [ G(e jθ ) I ] T [ γ 2 ][ G(e jθ ] ) < 0, θ Θ I The Q-filter has to be selected to cut-off the frequencies for which the stability condition is not satisfied.

110 GKYP ILC design A Nyquist based interpretation Im 1 Im 1 1 Re 1 Re j G(e ) j G(e ) σ max ( G(e jθ ) ) < 1 σ max ( G(e jθ ) ) > 1

111 Towards design procedures Let transfer-function product LS P have a state-space model realization defined by the following matrices [ ] [ ] Asp 0 Bsp A =, B =, B l C sp A l B l D sp where C = [ D l C sp C l ], D = Dl D sp A sp, B sp, C sp and D sp are a given state-space realization for S P and A l, B l, C l and D l are a state-space realization for L to be obtained.

112 Towards design procedures The LMI for L-filter design is obtained by application of the projection lemma and some standard matrix transformations The constraint that limits the solution to stable L must be added. The resulting L is stable and full-order and guarantees the ILC convergence condition The frequency ϖ l can be maximized by an optimization procedure Finally, the Q-filter can be chosen as a low-pass filter with cut-off frequency ϖ l.

113 Repetitive process based design Consider the following repetitive process description of an ILC law design problem [ ] [ ] [ ] ηk (p + 1) A BK = 1 BK 2 ηk (p) e k+1 (p) (CA + CBK 1 ) I CBK 2 e k (p) or, on introducing, A=A BK 1, B 0 =BK 2 C = (CA + CBK 1 ), D 0 = I CBK 2 [ ηk (p + 1) e k+1 (p) ] [ A B0 = C D 0 ] [ ηk (p) e k (p) ]

114 Practical example Consider the X-axis of the gantry robot. Then applying the design procedure for a cut-off frequency ϖ l = 15 Hz gives the following filter L state-space model matrices A l = Bl T = [ ] C l = [ ] D l =

115 Practical example - cont d The Q-filter is taken as a zero-phase Butterworth low-pass filter with cutoff frequency 15 Hz H(z) = z z z z z z 3 To implement this zero-phase filter the forward-backwards filtering method can be used, for which the Matlab command is filtfilt.m.

116 Practical example - cont d Frequency responses of the learning filter and process sensitivity function Frequency response of 1 L*S p L S p Magnitude (db) Magnitude (abs) Frequency (rad/sec) Frequency (rad/sec) Frequency responses of the learning filter and sensitivity function. Frequency response of 1 LS P.

117 Simulation result Convergence of the trial-to-trial tracking error is achieved RMS on each trial 10 2 RMS on each trial RMS(e) [m] 10 4 RMS(e) [m] the trial number the trial number After 10 trials After 100 trials

118 Further ILC design results The following result from repetitive process theory is used. Theorem A SISO discrete linear repetitive process, with the pair {A, B 0 } controllable and the pair {C, A} observable, is stable along the pass if and only if there exist R > 0, S > 0, Q > 0 and a symmetric matrix P such that the following LMIs are feasible i) D T 0 RD 0 R < 10 (in the SISO case D 0 < 1) ii) A T SA S < 0 iii) AP A T P QA T AQ+2Q AP C T QC T B 0 CP A T CQ CP C T 1 D 0 <0 B0 T D0 T 1

119 Finite frequency specification Theorem Consider an ILC scheme that can be written as a discrete linear repetitive process of the form considered here. Then stability along the trial holds over the finite frequency range, that is, the ILC scheme converges, if there exist S > 0, Q > 0 X 1, X 2, W, a symmetric matrix P, and scalars ρ 1, ρ 2 such that the following LMIs are feasible [ 1 CBX 2 X T 2 B T C T 2CBX 2 ] <0 [ S ρ1 W ρ 1 W T ρ 1 AW +ρ 1 BX 1 ρ 2 W T ( ) P Q+W 0 0 ( ) Ξ BX 2 W T A T C T X T 1 B T C T ( ) ( ) 1 X T 2 B T C T +1 ( ) ( ) ( ) 1 Γ ] <0 <0

120 ILC Design where Ξ =P 2 cos(ϖ l )Q sym(aw +BX 1 ) Γ = S + sym(ρ 2 AW + ρ 2 BX 1 ) ρ 1 ρ 2 +ρ 2 ρ 1 < 0 If these LMIs are feasible, control law matrices K 1 and K 2 can be calculated using K 1 =X 1 W 1, K 2 =X 2

121 Robust ILC Consider the case when { xk+1 (p + 1) = (A+ A)x P k+1 (p)+(b+ B)u k+1 (p) y k+1 (p) = Cx k+1 (p) The matrices A, B represent admissible uncertainties which are assumed to be of the (norm bounded) form A = HF E 1, B = HF E 2 where H, E 1, E 2 are known real constant matrices and F is an uncertain perturbation satisfying F F T I

122 Robust ILC Theorem Consider an ILC schemes which can be written as a discrete linear repetitive process with the norm bounded uncertainty structure considered above. Then this process is stable along the trial over the finite frequency range, i.e., the ILC scheme converges, if there exist matrices S > 0, Q > 0 X 1, X 2, W, a symmetric matrix P, and scalars ρ 1, ρ 2, ɛ 1 > 0, ɛ 2 > 0 and ɛ 3 > 0 such that the following LMIs are feasible 1+ɛCHH T C T CBX 2 ɛchh T C T 0 ( ) 2CBX 2 +ɛ 1 CHH T C T X T 2 E T 2 ( ) ( ) ɛ 1 S sym(ρ 1 W )+ɛ 2 ρ 2 1HH T Γ 2 0 ( ) Γ 1 W T E T 1 +X T 1 E T 2 ( ) ( ) ɛ 2 <0 <0

123 P Q+W ( ) Ξ + ɛ 3 HH T BX 2 W T A T C T X1 T BT C T X1 T ET 2 W T E1 T 0 ( ) ( ) 1 X2 T BT C T +1 X2 T ET 2 X2 T ET 2 ( ) ( ) ( ) 1+ɛ 3 CHH T C 0 0 <0 ( ) ( ) ( ) ( ) ɛ 3 0 ( ) ( ) ( ) ( ) ( ) ɛ 3 where Ξ =P 2 cos(ϖ l )Q sym(aw +BX 1 ) Γ 1 = S+sym(ρ 2 AW +ρ 2 BX 1 )+ɛ 2 ρ 2 2HH T Γ 2 =ρ 1 AW +ρ 1 BX 1 ρ 2 W T +ɛ 2 ρ 1 ρ 2 HH T ρ 1 ρ 2 +ρ 2 ρ 1 < 0 If these LMIs are feasible, control law matrices K 1 and K 2 can be calculated using K 1 =X 1 W 1, K 2 =X 2

124 Gantry robot example, cont d. This model is unstable along the trial. Take the uncertainty model matrices as E 1 = [ ] E 2 = 0.05 H = [ ] T

125 db Gantry robot example, cont d. Applying the design procedure for a cut-off frequency ϖ l = 10 Hz gives the control law matrices K 1 = [ ] K 2 = Nyquist plot of the designed ILC scheme. Nyquist Diagram 6 db 0.2 Imaginary Axis Real Axis

126 Practical example - cont d The Root Mean Square (RMS) values of the error for 100 trials is given below 10 2 RMS on each trial 10 3 RMS(e) [m] the trial number This simulation confirms that the designed ILC law stabilizes the system and results in convergence of the tracking error.

127 Zero Markov Parameters For most physical systems (e.g. mechanical) their transfer-function is strictly proper. Hence the degree of the denominator is greater than the degree of the numerator, e.g. G(s) = 2s + 3 s 3 + 2s 2 + 3s + 8 Let n denote the number of poles and m the number of zeros and set q = n m. Then in the state-space model ({A, B, C, D}) of such a system CA j B = 0 and D = 0, j = 0, 1,..., q In these notes attention is restricted to the case when CB = 0 but CAB 0.

128 Another variant of the KYP lemma Lemma For discrete linear time-invariant systems with G(e jθ )=C(e jθ I A) 1 B+D, θ [0, 2π] the following inequalities are equivalent (i) [ ] T [ ] I I G(e jθ Π ) G(e jθ < 0, θ [0, 2π] ) where Π is a given real symmetric matrix. (ii) HΩH T < 0 where Q > 0 and [ ] Ξ 0 Ω =, Ξ = 0 Π [ Q 0 0 Q ] [ ] I A 0 B, H = 0 C I D

129 Some LMI results Lemma The following statements are equivalent i) Let Φ, N and J be given and Φ + NJ T + JN T < 0, ii) Let Φ, N and J be given such as the LMI [ ] {[ ] Φ N F [J N T + sym T I ]} < 0 0 G is feasible in the matrix variables F and G.

130 New ILC design procedure Consider an ILC scheme written as a discrete linear repetitive process. Then stability along the trial, i.e., trial-to-trial error convergence, holds if there exist matrices Q > 0 N, K 2 such that the following LMIs are feasible [ I I CBK 2 I K2 T B T C T I [ Q N T B T + QA T AQ + BN Q ] <0 ] <0 Q ( ) ( ) ( ) AQ+BN Q ( ) ( ) CAQ CBN 0 I ( ) 0 K2 T B T I K2 T B T C T I < 0 (41) Also if these LMIs are feasible a stabilizing K 1 is given by K 1 =NQ 1 and a stabilizing K 2 can be computed directly from the decision matrices. (( ) denotes a block entry in a symmetric matrix).

131 Error signal filtering To remove the limitation arising from a case when CB = 0 (and hence no control at the first sample instant along the trial), the analysis that follows introduces a low-pass with unity dc-gain and sufficiently high cut-off frequency filter in series with ILC scheme and gives a design algorithm.. Suppose that the low-pass filter state-space model matrices are A f, B f, C f and D f. Then the state-space model matrices of the ILC law with this filter added in series is [ ] [ ] Af 0 Â=, B 0 C f A B Bf 0 = B 0 D f Ĉ= [ D 0 C f C ], D 0 = D 0 D f Also for the low-pass filter D f = 0 and hence D 0 = 0.

132 Error signal filtering For SISO systems the low-pass filter can be selected as first-order where A f, B f and C f are scalars and, for example, the observable canonical form state-space realization gives C f = 1. Moreover, the cut-off frequency has to be higher than the frequency spectrum of the signal to be tracked but not too high in order to attenuate high frequency noise. The LMI (41) applied to this case can be written as ( D 0 = 0) [ ÂQ Â T + B ] 0 BT 0 Q ÂQĈT < 0 ĈQÂT ĈQĈT I Also introduce the notation [ ] [ ] Af 0 0 A =, X = 0 A I K = [ ] [ K 2 K 1, C = Cf CA ]