Sizing of clock meshes ====================== Original by Lieven Vanderberghe. Adapted to CVX by Argyris Zymnis, 12/4/2005. Modified by Michael Grant, 3/8/2006. Adapted to CVXPY, with cosmetic modifications, by Judson Wilson, 5/26/2014. Topic References: - Section 4, L. Vandenberghe, S. Boyd, and A. El Gamal "Optimal Wire and Transistor Sizing for Circuits with Non-Tree Topology" Introduction ------------ We consider the problem of sizing a clock mesh, so as to minimize the total dissipated power under a constraint on the dominant time constant. The numbers of nodes in the mesh is :math:N per row or column (thus :math:n=(N+1)^2 in total). We divide the wire into m segments of width :math:x_i, :math:i = 1,\dots,m which is constrained as :math:0 \le x_i \le W_{\mbox{max}}. We use a pi-model of each wire segment, with capacitance :math:\beta_i x_i and conductance :math:\alpha_i x_i. Defining :math:C(x) = C_0+x_1 C_1 + x_2 C_ 2 + \cdots + x_m C_m we have that the dissipated power is equal to :math:\mathbf{1}^T C(x) \mathbf{1}. Thus to minimize the dissipated power subject to a constraint in the widths and a constraint in the dominant time constant, we solve the SDP .. math:: \begin{array}{ll} \mbox{minimize} & \mathbf{1}^T C(x) \mathbf{1} \\ \mbox{subject to} & T_{\mbox{max}} G(x) - C(x) \succeq 0 \\ & 0 \le x_i \le W_{\mbox{max}}. \end{array} Import and setup packages ------------------------- .. code:: python import cvxpy as cp import numpy as np import scipy as scipy import matplotlib.pyplot as plt # Show plots inline in ipython. %matplotlib inline # Plot properties. plt.rc('text', usetex=True) plt.rc('font', family='serif') font = {'weight' : 'normal', 'size' : 16} plt.rc('font', **font) Helper functions ---------------- .. code:: python # Computes the step response of a linear system. def simple_step(A, B, DT, N): n = A.shape[0] Ad = scipy.linalg.expm((A * DT)) Bd = (Ad - np.eye(n)).dot(B) Bd = np.linalg.solve(A, Bd) X = np.zeros((n, N)) for k in range(1, N): X[:, k] = Ad.dot(X[:, k-1]) + Bd; return X Generate problem data --------------------- .. code:: python # # Circuit parameters. # dim=4 # Grid is dimxdim (assume dim is even). n=(dim+1)**2 # Number of nodes. m=2*dim*(dim+1) # Number of wires. # 0 ... dim(dim+1)-1 are horizontal segments # (numbered rowwise); # dim(dim+1) ... 2*dim(dim+1)-1 are vertical # (numbered columnwise) beta = 0.5 # Capacitance per segment is twice beta times xi. alpha = 1 # Conductance per segment is alpha times xi. G0 = 1 # Source conductance. C0 = np.array([ ( 10, 2, 7, 5, 3), ( 8, 3, 9, 5, 5), ( 1, 8, 4, 9, 3), ( 7, 3, 6, 8, 2), ( 5, 2, 1, 9, 10) ]) wmax = 1 # Upper bound on x. # # Build capacitance and conductance matrices. # CC = np.zeros((dim+1, dim+1, dim+1, dim+1, m+1)) GG = np.zeros((dim+1, dim+1, dim+1, dim+1, m+1)) # Constant terms. # - Reshape order='F' is fortran order to match original # version in MATLAB code. CC[:, :, :, :, 0] = np.diag(C0.flatten(order='F')).reshape(dim+1, dim+1, dim+1, dim+1, order='F').copy() zo13 = np.zeros((2, 1, 2, 1)) zo13[:,0,:,0] = np.array([(1, 0), (0, 1)]) zo24 = np.zeros((1, 2, 1, 2)) zo24[0,:,0,:] = zo13[:, 0, :, 0] pn13 = np.zeros((2, 1, 2, 1)) pn13[:,0,:,0] = np.array([[1, -1], [-1, 1]]) pn24 = np.zeros((1, 2, 1, 2)) pn24[0, :, 0, :] = pn13[:, 0, :, 0] for i in range(dim+1): # Source conductance. # First driver in the middle of row 1. GG[int(dim/2), i, int(dim/2), i, 0] = G0 for j in range(dim): # Horizontal segments. node = 1 + j + i * dim CC[j:j+2, i, j:j+2, i, node] = beta * zo13[:, 0, :, 0] GG[j:j+2, i, j:j+2, i, node] = alpha * pn13[:, 0, :, 0] # Vertical segments. node = node + dim * ( dim + 1 ) CC[i, j:j+2, i, j:j+2, node] = beta * zo24[0, :, 0, :] GG[i, j:j+2, i, j:j+2, node] = alpha * pn24[0, :, 0, :] # Reshape for ease of use. CC = CC.reshape((n*n, m+1), order='F').copy() GG = GG.reshape((n*n, m+1), order='F').copy() # # Compute points the tradeoff curve, and the three sample points. # npts = 50 delays = np.linspace(50, 150, npts) xdelays = [50, 100] xnpts = len(xdelays) areas = np.zeros(npts) xareas = dict() Solve problem and display results --------------------------------- .. code:: python # Iterate over all points, and revisit specific points for i in range(npts + xnpts): # First pass, only gather minimal data from all cases. if i < npts: delay = delays[i] print( ('Point {} of {} on the tradeoff curve ' \ + '(Tmax = {})').format(i+1, npts, delay)) # Second pass, gather more data for specific cases, # and make plots (later). else: xi = i - npts delay = xdelays[xi] print( ('Particular solution {} of {} ' \ + '(Tmax = {})').format(xi+1, xnpts, delay)) # # Construct and solve the convex model. # # Variables. xt = cp.Variable(shape=(m+1)) # Element 1 of xt == 1 below. G = cp.Variable((n,n), symmetric=True) # Symmetric constraint below. C = cp.Variable((n,n), symmetric=True) # Symmetric constraint below. # Objective. obj = cp.Minimize(cp.sum(C)) # Constraints. constraints = [ xt[0] == 1, G == G.T, C == C.T, G == cp.reshape(GG*xt, (n,n)), C == cp.reshape(CC*xt, (n,n)), delay * G - C == cp.Variable(shape=(n,n), PSD=True), 0 <= xt[1:], xt[1:] <= wmax, ] # Solve problem (use CVXOPT instead of SCS to match original results; # cvxopt produces lower objective values as well, but is much slower) prob = cp.Problem(obj, constraints) try: prob.solve(solver=cp.CVXOPT) except cp.SolverError: print("CVXOPT failed, trying robust KKT") prob.solve(solver=cp.CVXOPT, kktsolver='robust') if prob.status not in [cp.OPTIMAL, cp.OPTIMAL_INACCURATE]: raise Exception('CVXPY Error') # Chop off the first element of x, which is # constrainted to be 1 x = xt.value[1:] # First pass, only gather minimal data from all cases. if i < npts: areas[i] = sum(x) # Second pass, gather more data for specific cases, # and make plots. else: xareas[xi] = sum(x) # # Print display sizes. # print('Solution {}:'.format(xi+1)) print('Vertical segments:') print(x[0:dim*(dim+1)].reshape(dim, dim+1, order='F').copy()) print('Horizontal segments:') print(x[dim*(dim+1):].reshape(dim, dim+1, order='F').copy()) # # Determine and plot the step responses. # A = -np.linalg.inv(C.value).dot(G.value) B = -A.dot(np.ones(n)) T = np.linspace(0, 500, 2000) Y = simple_step(A, B, T[1], len(T)) indmax = -1 indmin = np.inf for j in range(Y.shape[0]): inds = np.amin(np.nonzero(Y[j, :] >= 0.5)[0]) if ( inds > indmax ): indmax = inds jmax = j if ( inds < indmin ): indmin = inds jmin = j tthres = T[indmax] GinvC = np.linalg.solve(G.value, C.value) tdom = max(np.linalg.eig(GinvC)[0]) elmore = np.amax(np.sum(GinvC.T, 0)) plt.figure(figsize=(8, 8)) plt.plot( T, np.asarray(Y[jmax,:]).flatten(), '-', T, np.asarray(Y[jmin,:]).flatten() ) plt.plot( tdom * np.array([1, 1]), [0, 1], '--', elmore * np.array([1, 1]), [0, 1], '--', tthres * np.array([1, 1]), [0, 1], '--' ) plt.xlim([0, 500]) plt.ylim([0, 1]) plt.text(tdom, 0.92, 'd') plt.text(elmore, 0.88, 'e') plt.text(tthres, 0.96, 't') plt.text( T[600], Y[jmax, 600], 'v{}'.format(jmax+1)) plt.text( T[600], Y[jmin, 600], 'v{}'.format(jmin+1)) plt.title(('Solution {} (Tmax={}), fastest ' \ + 'and slowest step responses').format(xi+1, delay), fontsize=16) plt.show() # # Plot the tradeoff curve. # plt.figure(figsize=(8, 8)) ind = np.isfinite(areas) plt.plot(areas[ind], delays[ind]) plt.xlabel('Area') plt.ylabel('Tdom') plt.title('Area-delay tradeoff curve') # Label the specific cases. for k in range(xnpts): plt.text(xareas[k], xdelays[k], '({})'.format(k+1)) plt.show() .. parsed-literal:: Point 1 of 50 on the tradeoff curve (Tmax = 50.0) CVXOPT failed, trying robust KKT Point 2 of 50 on the tradeoff curve (Tmax = 52.04081632653061) Point 3 of 50 on the tradeoff curve (Tmax = 54.08163265306123) Point 4 of 50 on the tradeoff curve (Tmax = 56.12244897959184) Point 5 of 50 on the tradeoff curve (Tmax = 58.16326530612245) Point 6 of 50 on the tradeoff curve (Tmax = 60.20408163265306) Point 7 of 50 on the tradeoff curve (Tmax = 62.244897959183675) Point 8 of 50 on the tradeoff curve (Tmax = 64.28571428571429) Point 9 of 50 on the tradeoff curve (Tmax = 66.3265306122449) Point 10 of 50 on the tradeoff curve (Tmax = 68.36734693877551) Point 11 of 50 on the tradeoff curve (Tmax = 70.40816326530611) Point 12 of 50 on the tradeoff curve (Tmax = 72.44897959183673) Point 13 of 50 on the tradeoff curve (Tmax = 74.48979591836735) Point 14 of 50 on the tradeoff curve (Tmax = 76.53061224489795) Point 15 of 50 on the tradeoff curve (Tmax = 78.57142857142857) Point 16 of 50 on the tradeoff curve (Tmax = 80.61224489795919) Point 17 of 50 on the tradeoff curve (Tmax = 82.65306122448979) Point 18 of 50 on the tradeoff curve (Tmax = 84.6938775510204) Point 19 of 50 on the tradeoff curve (Tmax = 86.73469387755102) Point 20 of 50 on the tradeoff curve (Tmax = 88.77551020408163) Point 21 of 50 on the tradeoff curve (Tmax = 90.81632653061224) Point 22 of 50 on the tradeoff curve (Tmax = 92.85714285714286) Point 23 of 50 on the tradeoff curve (Tmax = 94.89795918367346) Point 24 of 50 on the tradeoff curve (Tmax = 96.93877551020408) Point 25 of 50 on the tradeoff curve (Tmax = 98.9795918367347) Point 26 of 50 on the tradeoff curve (Tmax = 101.0204081632653) Point 27 of 50 on the tradeoff curve (Tmax = 103.06122448979592) Point 28 of 50 on the tradeoff curve (Tmax = 105.10204081632654) Point 29 of 50 on the tradeoff curve (Tmax = 107.14285714285714) Point 30 of 50 on the tradeoff curve (Tmax = 109.18367346938776) Point 31 of 50 on the tradeoff curve (Tmax = 111.22448979591837) Point 32 of 50 on the tradeoff curve (Tmax = 113.26530612244898) Point 33 of 50 on the tradeoff curve (Tmax = 115.3061224489796) Point 34 of 50 on the tradeoff curve (Tmax = 117.34693877551021) Point 35 of 50 on the tradeoff curve (Tmax = 119.38775510204081) Point 36 of 50 on the tradeoff curve (Tmax = 121.42857142857143) Point 37 of 50 on the tradeoff curve (Tmax = 123.46938775510205) Point 38 of 50 on the tradeoff curve (Tmax = 125.51020408163265) Point 39 of 50 on the tradeoff curve (Tmax = 127.55102040816327) Point 40 of 50 on the tradeoff curve (Tmax = 129.59183673469389) Point 41 of 50 on the tradeoff curve (Tmax = 131.6326530612245) Point 42 of 50 on the tradeoff curve (Tmax = 133.67346938775512) Point 43 of 50 on the tradeoff curve (Tmax = 135.71428571428572) Point 44 of 50 on the tradeoff curve (Tmax = 137.75510204081633) Point 45 of 50 on the tradeoff curve (Tmax = 139.79591836734693) Point 46 of 50 on the tradeoff curve (Tmax = 141.83673469387756) Point 47 of 50 on the tradeoff curve (Tmax = 143.87755102040816) Point 48 of 50 on the tradeoff curve (Tmax = 145.9183673469388) Point 49 of 50 on the tradeoff curve (Tmax = 147.9591836734694) Point 50 of 50 on the tradeoff curve (Tmax = 150.0) Particular solution 1 of 2 (Tmax = 50) CVXOPT failed, trying robust KKT Solution 1: Vertical segments: [[0.65284441 0.4391586 0.52378143 0.47092764 0.2363529 ] [0.99999993 0.85353862 0.99999992 0.93601078 0.56994586] [0.92325575 0.29557654 0.80041338 0.99999998 0.99999997] [0.41300012 0.13553757 0.26699524 0.67049218 0.88916807]] Horizontal segments: [[1.96487539e-01 1.40591789e-01 9.70591442e-08 7.79376843e-08 5.27429285e-08] [7.07446433e-02 6.38430105e-02 1.02136471e-07 8.59913722e-08 6.28906472e-08] [6.05807467e-09 1.16285450e-08 3.91561390e-08 9.48052913e-02 1.58096913e-01] [3.82528741e-07 4.85708568e-07 5.75578696e-07 8.39862772e-02 5.38639181e-02]] .. image:: clock_mesh_files/clock_mesh_7_1.png .. parsed-literal:: Particular solution 2 of 2 (Tmax = 100) Solution 2: Vertical segments: [[0.2687881 0.04368684 0.17122095 0.133796 0.07360396] [0.41346231 0.08016135 0.30642705 0.2224136 0.1484946 ] [0.25755998 0.08016077 0.11200259 0.38352317 0.28159768] [0.13439419 0.04368697 0.02445701 0.24083502 0.24534599]] Horizontal segments: [[ 1.53896782e-09 -5.18600578e-10 -9.75218556e-10 -5.19196383e-10 1.57176577e-09] [ 9.30752726e-10 -9.56673760e-10 -1.35065528e-09 -9.96797753e-10 1.03852376e-09] [ 9.35404466e-10 -9.12219313e-10 -2.22938358e-10 -7.91865186e-10 1.51304362e-09] [ 1.31975762e-09 -8.50790152e-10 -1.39421076e-09 -8.33247519e-10 1.27680128e-09]] .. image:: clock_mesh_files/clock_mesh_7_3.png .. image:: clock_mesh_files/clock_mesh_7_4.png