Clock Synchronization Errors: The robustness problem

Transcription

1 Clock Synchronization Errors: The robustness problem Marek Przedwojski 1 Supervisors: Professor Eric Rogers 1, Dr Ivan Markovsky 1 1 University of Southampton, United Kingdom BELS Control Meeting in Exeter, January 2011

2 Outline 1 Problem description Model Main stability results 2 3

3 What we consider? Problem description Model Main stability results Digital LTI systems that Consist of subsystems Subsystems change their state (switches) at certain time instances defined by rising or falling edge of the clock signal If a subsytems switches Then the corrensponding state vector entries are recalculated and take new values

4 What is a clock synchronization error? Problem description Model Main stability results Synchronization errors Clock synchronization errors occurs if the subsystems switches at different point of time. The possible reasons are Signal propagation delays, i.e. clock signal reaches the subsystems at different points of time Subsystems are driven by clocks with different frequencies Definition A clock synchronization error is the error in system s response due to nonsynchronous switching of subsystems.

5 Common clock case Problem description Model Main stability results Figure: The common clock case. Large distance causes signal propagation delays and phase shifts. Distributed systems.

6 Different clocks case Problem description Model Main stability results Figure: The case with different clocks and frequencies. Interconnected systems or swarms.

7 Typical examples Problem description Model Main stability results Typical examples include High clock speed circuitry (due to signal propagation delays) Distributed systems Sensor networks Multi-agent systems, swarms Interconnected systems

8 Synchronous equation Problem description Model Main stability results Synchronous system All subsystems switch at the same time points State vector obeys x(k + 1) = Ax(k) + Bu(k) (1) for some matrices A, B and u as system input System with a clock synchronization error Subsystems switch at different points of time Equation (1) is no longer valid

9 An Example Problem description Model Main stability results Assume System consist of two subsystems with corresponding entries x 1 and x 2 respectively Sequence s = ({1}, {2}) describes switching event pattern which means that x 1 updates always before x 2 (due to phase clock shifts) In case of no synchronization errors system would update according to [ x1 x 2 ] [ a11 a (k + 1) = 12 a 21 a 22 ] [ x1 x 2 [ b11 b + 12 b 21 b 22 ] (k) ] [ u1 u 2 ] (k) (2)

10 An Example - updating of a system Problem description Model Main stability results x 1 updates: [ x 1 1 x 1 2 ] [ (k) = a 11 a ] [ x 1 x 2 (k)+ ] [ = A {1} [ x1 x 2 b 11 b ] [ ] (k) + B {1} [ u1 u 2 ] u 1 (k) u 2 ] (k) x 2 updates: [ ] [ x 1 (k+1) = x 2 ] [ 1 0 a 21 a 22 x 1 1 x 1 2 = A {2} [ x 1 1 x 1 2 ] [ (k)+ ] [ 0 0 b 21 b 22 ] (k) + B {2} [ u1 u 2 ] (k) ] u 1 (k) u 2

11 An Example - State transition Problem description Model Main stability results By back-substitution the state transition can be writen [ x1 x 2 ] (k + 1) = A s [ x1 x 2 ] (k) + B s [ u1 u 2 ] (k) where A s = A {2} A {1}, B s = ( B {2} + A {2} B {1} ) This is not original system and sometimes some stable systems may become unstable

12 Problem description Problem description Model Main stability results The robustness problem in general Find a suitable controller able to stabilize plant in case of every clock synchronization error.

13 to the model Problem description Model Main stability results Seqence s In general sequence of sets of indices s = (i 1,..., i d ) describes the order in which variables update and which variables update simultaneously. E.g, for three order system s = ({2, 3}, {1}) means that variables x 2, x 3 update simultaneously before x 1. For n order system the number of all possible sequences can be approximated by 2 n n! in a common clock case. Definition The sequence s describing switching event pattern we call a synchronization error

14 Model Problem description Model Main stability results Updating event When the jth event occurs state vector updates according to x j (k) = A ij x j 1 (k) + B ij u(k), j = 1,..., d (3) The model matrices A ij and B ij for e.g. given i j = {..., p,..., q,...} are A ij = a p1 a p2... a pn a q1 a q2... a qn , B ij = b p1 b p2... b pm b q1 b q2... b qm (4)

15 State transition Problem description Model Main stability results For the sequence s = (i 1,..., i d ) state vector updates in d steps x 0 (k) = x(k)... =... x j (k) = A ij x j 1 (k) + B ij u(k)... =... x(k + 1) = x d (k) Let A s = A id A i1 B s = B id + A id B id A id A i2 B i1 And by back-substitution we obtain x(k + 1) = A s x(k) + B s u(k) (5)

16 Asynchronous algorithms Problem description Model Main stability results Consider the iterative algorithm x(p + 1) = Ax(p) (6) Now assume that n processors calculate n state vector entries separately and they share the memory. That algorithm is called the asynchronous algorithm and the equation becomes x(p + 1) = A i(p) x(p), i(p) {1,..., n} (7) where for e.g. given i(p) = {..., p,..., q,...} are A i(p) = a p1 a p2... a pn a q1 a q2... a qn (8)

17 Asynchronous algorithms Problem description Model Main stability results Definition The sequence i(p), p = 0, 1,... is admissible if every i {1,..., n} belongs to infitely many subsets which means every state vector entry updates infinitely many often. Relation to clock synchronization errors The evolution of an autonomous system with clock synchronization error is the example of asynchronous algorithm with an admissible sequence.

18 Stability conditions Problem description Model Main stability results Chasan and Miranker 1969 The asynchronous algorithm is asymptoticaly stable in the class of all admissible sequences if and only if Asarin, Kozyakin et al 1992 ρ( A ) < 1 (9) If the matrix is symmetric A = A T then the asynchronous algorithm is asymptotically stable in the class of all admissible sequences if and only if ρ(a) < 1 (10)

19 Application to the robustness problem Problem description Model Main stability results Problem Both stability conditions in order to use require the controller to update the input immediately after the updating of state vector entry. It is impossible from the engineering point of view because controller is driven by its own clock. Solution We need to develop other methods to deal with a problem.

20 Other work Problem description Model Main stability results Previous work The first engineering approach was taken in [1] The model was presented in [3] along with a method of identification of clock synchronization error. Stability of clock synchronization errors is studied in [2] (including cases with different clock frequencies). Main references 1 A.F. Kleptysyn, V.S. Kozyakin, M.A. Krasnoselski and N.A. Kuzntesov, Effect of small synchronization errors on stability of complex systems, C. Lorand, Theory of synchronization errors,ph.d.thesis, C. Lorand, P. Bauer, Clock synchronization errors in circuits: Model, Stability and Fault Detection, 2006

21 Stabilization Set of all matrices Let S be the set of all sequences s for the given n-order system. Let A = {[A s B s ] : s S} (11) card(a) 2 n n! The problem we solve To produce state feedback controller (u = K x) able to stabilize the system in case of all clock synchronization errors, i.e., such that closed loop matrix A s + B s K is a stable matrix for all s S.

22 Direct computation Feasibility problem The system is quadratically stabilizable if and only if there exist matrices Q = Q T and R such that [ ] Q A s Q + B s R QA T s + R T Bs T 0, s S (12) Q where 0 means negative definite. The state feedback controller is given by K = RQ 1 (13) Corollary We may use different target like H 2 /H optimal control.

23 The problem The problem Computation becomes very intensive for larger n. The solution Compute bounding polytope P for the set A with lower number of vertices. Then solve feasibility problem for vertices of the polytope in order to obtain a controller. This way should be faster than the direct computation and offer good accuracy. Algorithm Existing tools like qhull and GBT were too slow to be usable for that problem. We invented very fast algorithm to compute the approximate bounding polytope that offer the desired accuracy.

24 The idea Main idea The idea was to compute Minimum Volume Enclosing Ellipsoid (MVEE) for the Matrices in a vector space. Then to compute the approximate polytope with low number of vertices. MVEE For the computation of MVEE we used the modified version of the Khachiyan algorithm [1] implemented in Geometric Bounding Toolbox library v7. 1. L. G. Khachiyan Rounding polytopes in the real number model of computation, Math. of Op. Research 1996

25 Observation Figure: The observation is that all the points lie always on d dimensional hyperplane. By translation we obtain the d dimensional subspace. Then we calculate MVEE.

26 Algorithm -step 5 Visualization By changing the basis MVEE is transformed into a ball of unit radius. Vertices are chosen on each axes such that the convex hull contains the entrie ball in R d dimensional space M. Przedwojski Stability and robustness 3 of systems with synchronization errors

27 Algorithm Steps of the algorithm: 1 Map matrices onto vectors 2 Reduce the dimension to a subspace spanned 3 Compute the minimum volume ellipsoid containing all the vectors 4 Compute the transformation that maps the ellipsoid into a ball of unit radius 5 Choose vertices of the bounding polytope using symmetry of the ball 6 By back-transformation obtain the vertices as matrices

28 Performance Comparison of direct computation vs. computation with polytope approximation - Over 1 hour direct computation n computation with new algorithm avg time (sec) avg time(sec) * 6000

29 Norm bounded uncertainty Norm bounded uncertainty x(k + 1) = Ax(k) + Hp(k) + Bu(k), x(0) = x 0 q(k) = E 1 x(k) + E 2 u(k) y(k) = Cx(k) + Du(k) p(k) = F (k)q(k), F (k) 1 which may be rewritten as x(k + 1) = (A + HF (k)e 1 )x(k) + (B + HF (k)e 2 )u(k), x(0) = y(k) = Cx(k) + Du(k), F (k) 1 Remark One single LMI condition is needed to solve in order to find a controller.

30 Problem statement The problem Assume we have a set of matrices [ A k, B k ], k = 1,..., N The problem is to find the minimum volume norm bounded uncertainty containing all the matrices. Formulation minimize overa,b,h,e µ({hf E : F T F I}) subject to [A k B k ] = [A B] + HF k E, F T k F k I, k = 1,..., N

31 Proposition It is known that F 2 F F, with F denoting the Frobenius norm. Consider the sets Ω = {[A B]+HF E : F 2 1}, Ω F = {[A B]+HF E : F F 1}, Define the vec operation that maps given matrix into a vector m m 1n vec(m) = vec... = m m1... m mn m 11. m m1 m 12. m mn R mn

32 Norm bounded uncertainty as ellipsoid We establish now the image Ω of the set Ω f under the vec operation Ω = {vec([a B]) + E T H vec(f ) : vec(f ) 2 1} Let The set a = vec([a B]), f = vec(f ), R = E T H (14) Ω = {a + Rf : f 2 1} (15) represents an ellipsoid. Assume n v = dim(a), d v = dim(f ), e v = rank(r) In general, we have e v dimensional ellipsod in the n v = n(n + m) dimensional space as the image of the d v dimensional ball.

33 Uncertainty to ellipsoid relation Figure: Uncertainty defined with the Frobenius norm corresponds to the ellipsoid.

34 The measure of the uncertainty The length of axes of the ellipsoid is given by the singular values σ 1,..., σ ev of the matrix R. The volume is proportional to the product of the nonzero singular values. Due to the fact that usually e v is large we take the log of the value as the measure in order to avoid large numbers. Thus we set µ({ HF E : F T F I }) = log σ 1 σ ev (16) where σ 1,..., σ ev are singular values of E T H.

35 Problem formulation Thus the original problem may be stated as follows minimize overa,b,h,e log σ 1 σ ev subject to σ 1,..., σ ev are all nonzero singular values of E T H [A k B k ] = [A B] + HF k E, F T k F k I, k = 1,..., N However, in our approach we solve the related problem with the Frobenius norm minimize overa,b,h,e log σ 1 σ ev subject to σ 1,..., σ ev are all nonzero singular values of E T H [A k B k ] = [A B] + HF k E, F k F 1, k = 1,..., N

36 Taking x k = vec([a k B k ]) we obtain the problem formulated completely in the vector space minimize overa,b,h,e log σ 1 σ ev subject to σ 1,..., σ ev are all nonzero singular values of E T H and x k = a + (E T H)f k, f k 2 1 k = 1,..., N or (x k a) T (E T H) 2 (x k a) 1, k = 1,..., N The latest constraint comes from the different ways the ellipsoid may be represented. If we assume that the input points x k, k = 1,..., N span the whole space then the problem has a formulation minimize overa,b,h,e log det (E T H) subject to (x k a) T (E T H) 2 (x k a) 1, k = 1,..., N

37 Input points span the whole space We assume that the input points x k = vec([ A k B k ]), k = 1,..., N span the whole space. Then we solve the problem minimize overa,b,h,e log det (E T H) subject to (x k a) T (E T H) 2 (x k a) 1, k = 1,..., N a = vec([ A B ])

38 The idea again Figure: We compute the MVEE ellipsoid containing input points and replacing the constarints in the original problem. Then we are looking for the ellipsoid E T H containing the MVEE.

39 The method 1 Compute Minimum Volume Enclosing Ellipsoid (MVEE) containing the input points. Solve the problem minimize over P,a log det P subject to (x k a) T P 2 (x k a) 1, k = 1,..., N (17) 2 Using the MVEE ellipsoid obtain the ellipsoid that represents the norm bounded uncertainty. Approximate solution (which may not be optimal) is done in two steps 1 Solve the following problem minimize overe,h P E T H F (18) 2 Rescale the ellipsoid (E T H, a) to ensure it contains all the points.

40 Frobenius norm minimization The solution to problem 2.(a) is given in [2]. First, we obtain a solution using rank-1 approximation. Next, the solution is modified by solving a series of least squares problems. 1. L.G. Kachiyan, "Rounding of polytopes in the real number model of computation", Mathematics of Operations Research, Charles van Loan and Nikos Pitsianis,"Approximation with kronecker products", Lin. Alg. for Large Scale and Real Time Computations, 1993

41 Rescaling Assume that points z 1,..., z p lie outside the ellipsoid (R, a). First compute vectors f 1 = R 1 (z a),..., f p = R 1 (z a) and norms f k 2 > 1, k = 1,..., p for all the points. We take the largest norm, i.e. let n max = max k=1,...,p f k 2. Let R = n max R and observe that vectors f 1 = R 1 (z a),..., f p = R 1 (z a) have the norm less than one, i.e. f k 2 = f k 2 /n max 1, k = 1,..., p. That means all the points lie entirely inside the ellipsoid (R, a) = (n max R, a).

42 Input points an a hyperplane Figure: Recall that for systems with clock synchronization errors the input points lie on some d-dimensional hyperplane.

43 Wrong approximation in lower dimension Figure: Performing approximation in Frobenius norm for the lower dimensional ellipsoid may fail. The resulting ellipsoid may lie on different hyperplane and may do not contain points at all.

44 Lifting Figure: Lifting means adding to the MVEE small nonzero length semiaxes from perpendicular space obtaining fully dimensional ellipsoid.

45 Input points on a hyperplane 1 Compute Minimum Volume Enclosing Ellipsoid (MVEE) containing the input points. Solve the problem minimize over P,a log det P subject to (x k a) T P 2 (x k a) 1, k = 1,..., N (19) 2 Lift the ellipsoid to R nv 3 Using the lifted MVEE ellipsoid obtain the ellipsoid that represents the norm bounded uncertainty. Approximate solution (which may not be optimal) is done in two steps 1 Solve the following problem minimize overe,h P E T H F (20) 2 Rescale the ellipsoid (E T H, a) to ensure it contains all the points.

46 The matrix space method The method solves the following problem minimize overx,h,e trhh T + tre T E subject to X k = X + HF k E, F T k F k I, k = 1,..., N Where X k = [ A k, B k ], k = 1,..., N. The solution is obtained by solving a system of LMI. Details are given in [1]. [1] S. Boyd,"LMI in System and Control theory", p. 58

47 Comparison of the time of computation Number Matrix Heuristic Matrix Space/Heuristic of Space method time points method ratio Table: Average time of computation in seconds for 3rd-order system.

48 Time of computation - picture Figure: Matrix Space/Heuristic time ratio for 3rd-order system.

49 In other words Example For 1 million input points the proposed method will compute the uncertainty after 1 hour whereas the existing method after 11 years.

50 Tests In the following tests we compare two methods - The proposed Heuristic method with the existing Matrix Space method. In all tests we assume that the input points span the whole space. Define the factor det E T H LMF = n v det E T m H m where H, E describes the norm bounded uncertainty obtained by the Heuristic method and H m, E m are given by the Matrix space one. The LMF (Linear Magnification Factor) parameter determines the factor by which every exes of the ellipsoid E T m H m should be multiplied in order to have the same volume as the ellipsoid E T H.

51 The measure of the uncerainty Number Matrix space Heuristic Linear of uncertainty uncertainty magnification points measure measure Table: Comparison of the measures of the norm bounded uncertainty obtained by different methods for 3rd-order system.

52 Evolution of the measure Order Dimension Linear Matrix Space/Heuristic of a system of a space magnification time ratio Table: Evolution of the time ratio factor and the linear magnification factor with the dimension of the space

53 Conclusions Advantage The time. The heuristic method is c N, (N-number of input points) faster than the Matrix Space method. Disadvantage Volume redundancy. Future work If the stability conditions may be derived for the norm bounded uncertainty defined with the Frobenius norm then the volume redundancy would disappear.

54 An example Consider a swarm system that consisting of M agents each of which is modeled as [ ] [ ] ([ ] [ ] ) x1 δi ω (n + 1) = i x1 u1 (n) (n) (21) x 2 ω i δ i x 2 u 2 i where i = 1, 2,... M and The agents work together and aim to meet at the rendezvous point.the process input [ u 1 u 2 ] T i is defined as [ u1 u 2 ] (n) = 1 M ( M i=1 [ x1 x 2 ] i (n) i ) [ e1 + e 2 ] i i (n) (22) where [ e 1, e 2 ] T i, i = 1,..., M denotes an independent input of each agent.

55 An Example - Continued Each agent calculate the center of positions of all agents and move with rotation towards the center. If agents perform calculations at different points of time then the entire system can be represented as system with clock synchronization errors In the example we considered a system of 6 agents. This system was unstable for some synchronization error. Using presented algorithm we designed a state feedback controller that make the system stable.

56 Agents Figure: Agents aim to meet at the center of position of all agents. If they operate synchronously they calculate the same target point and finally meet.

57 Agents with clock synchronization errors Figure: If they calculate the center at different point of time they run towards different points.

58 Time of computation - picture Figure: Stabilizing the system we allowed agents to meet.

59 Thank you!

60 Appendix Algorithm Algorithm - step 1 Mapping matrices onto vectors M = [ m ij ] 1 i n,1 j m x = [ x k ] 1 k m n Define the invertible map φ : M m n (R) R m n as and x = φ(m) = [ m 11, m 21,..., m m,1, m 12, m 22,..., m m,2,......, m 1n, m 2n,... m m,n ] (23) M = φ 1 (x) = x 1 x m+1... x (n 1) m+1 x 2 x m+2... x (n 1) m x m x 2m... x n m (24)

61 Appendix Algorithm Algorithm - step 2 Translate set of all vectors P of c = 1 n p np i=1 x i P(c) = {x R n2 +n l : x c P} (25) where n p is the number of vectors Calculate the orthonormal basis of subspace spanned B = [ b 1, b 2,..., b d ] (26) Calculate new coordinates x = Bx x = B T x (27)

62 Appendix Algorithm Algorithm - step 3 Compute minimum volume ellipsoid containing all the vectors x E = {y R d : (y e) T E(y e) 1} (28) We used an algorithm implemented in GBT 7.0

63 Appendix Algorithm Algorithm - step 4 Using Cholesky factorization obtain a matrix H such that E = H T H Now let z = Hy, f = He. This transformation changes the entire ellipsoid into a ball of unit radius H(E ) = {z R d : (z f) T (z f) 1} = B(f, 1) (29) Back-transformation is defined as y = H 1 z

64 Appendix Algorithm Algorithm - step 5 Chose vertices as d 11 = [ k, 0, 0,..., 0] T + f d d1 = [ 0, 0, 0,..., k ] T + f d 12 = [ k, 0, 0,..., 0 ] T + f d d2 = [ 0, 0, 0,..., k ] T + f where k = d (d-dimension of the space). Then it is guaranteed that the convex hull contains the entire ball Number of the vertices equals twice the dimension and it is a polynomial function of the order of the system

65 Appendix Algorithm Algorithm - step 6 Obtaining vertices d ij as matrices involve: Transform coordinates using H 1 Transform coordinates suing B Translate of c Map coordinates to matrices All the operations are linear and the convex hull remains the convex hull.

66 Appendix Algorithm The measure of the norm bounded uncertainty First proposal 1 The volume of the uncertainty in the matrix space. But matrices may lie on a hyperplane and the volume is zero in that case. 2 The diameter of the set max { F G : F, G Ω} = 2 λ max (HH T )λ max (E T E) There may exist two uncerainties of the same measure and one may be a subset of the other. The problem is nonunique then.