├── LICENSE ├── README.md ├── accSimpleRmpcLpv.py ├── disturbanceFeedbackRmpcLti.py ├── optimizedTighteningRmpcLpv.py ├── problemDef.py ├── rigidTubeMpcLti.py ├── setOperations.py ├── shrinkingTubeMpcLti.py └── solveMPC.py /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # RMPCPy 2 | Python code for implementing a set of basic robust model predictive control (RMPC) algorithms for linear systems. The algorithms incorporated in this repository are for both linear time-invariant (LTI) and linear parameter-varying (LPV) systems. These algorithms are listed below: 3 | 4 | 1. **Shrinking Tube MPC for LTI Systems:** Chisci L, Rossiter JA, Zappa G. _Systems with persistent disturbances: predictive control with restricted constraints._ Automatica 2001. Available at: https://www.sciencedirect.com/science/article/pii/S0005109801000516 5 | 6 | 2. **Rigid Tube MPC for LTI Systems:** Mayne DQ, Seron MM, Raković SV. _Robust model predictive control of constrained linear systems with bounded disturbances._ Automatica 2005. Available at: https://www.sciencedirect.com/science/article/pii/S0005109804002870 7 | 8 | 3. **Disturbance Feedback Robust MPC for LTI Systems:** Goulart PJ, Kerrigan EC, Maciejowski JM. _Optimization over state feedback policies for robust control with constraints._ Automatica 2006. Available at: https://www.sciencedirect.com/science/article/pii/S0005109806000021?via%3Dihub 9 | 10 | 4. **A Simple Robust MPC for LPV Systems:** Bujarbaruah M, Rosolia U, Stürz YR, Borrelli F. _A simple robust MPC for linear systems with parametric and additive uncertainty._ IEEE American Control Conference 2021. Available at: https://ieeexplore.ieee.org/document/9482957 11 | 12 | 5. **Robust MPC for LPV Systems with Optimization-Based Constraint Tightening:** Bujarbaruah M, Rosolia U, Stürz YR, Zhang X, Borrelli F. _Robust MPC for linear systems with parametric and additive uncertainty: A novel constraint tightening approach._ arXiv preprint, 2020. Available at: https://arxiv.org/abs/2007.00930 13 | -------------------------------------------------------------------------------- /accSimpleRmpcLpv.py: -------------------------------------------------------------------------------- 1 | # Code implementing "A simple robust MPC for LPV systems". 2 | # From the paper: 3 | # 4 | # Bujarbaruah M, Rosolia U, Stürz YR, Borrelli F. A simple robust MPC for linear 5 | # systems with parametric and additive uncertainty. American Control Conference, 6 | # 2021, pp. 2108-2113, IEEE. 7 | # 8 | # This code does not do closed-loop MPC. It generates an inner approx. of the ROA. 9 | import numpy as np 10 | import problemDef as pdef 11 | import polytope as pc 12 | from solveMPC import solveMpcAccSimpleRobust 13 | import matplotlib.pyplot as plt 14 | 15 | # Load the parameters of the problem here. 16 | params = pdef.ProblemParams() 17 | 18 | # Assign the horizon of the MPC. 19 | params.setHorizon(5) 20 | 21 | # Assign the number of initial condition samples to use. 22 | params.setx0SampleCount(576) 23 | 24 | # Compute the terminal set Xn. 25 | params.computeMaxRobustPosInvariantLPV() 26 | 27 | # Compute the big matrices needed for all horizons in distFeedback. 28 | dictOfMatricesDf = [] 29 | for i in range(1,params.N+1): 30 | dictOfMatricesDf.append(params.formDfMatrices(i, "LPV")) 31 | 32 | # Solve MPC from a bunch of sampled initial conditions. 33 | # Store feasible initial conditions as ROA samples. 34 | xs = params.getInitialStateMesh() 35 | xFeas = np.zeros([1, params.nx]) 36 | 37 | # The matrices required for horizon 1 problem. 38 | boldAbar = params.Anom 39 | boldBbar = params.Bnom 40 | Fx = params.XnLPV.A 41 | fx = params.XnLPV.b 42 | boldHw = params.W.A 43 | boldhw = params.W.b 44 | boldHu = params.U.A 45 | boldhu = params.U.b 46 | 47 | dictofMatricesH1 = dict(boldAbar=boldAbar, 48 | boldBbar=boldBbar, 49 | Fx=Fx, 50 | fx=fx, 51 | boldHw=boldHw, 52 | boldhw=boldhw, 53 | boldHu=boldHu, 54 | boldhu=boldhu) 55 | 56 | for j in range(params.Nx): 57 | solverSuccessFlag = False 58 | for i in range(1,params.N+1): 59 | solverSuccessFlag += solveMpcAccSimpleRobust(xs[:,j], 60 | params, 61 | dictOfMatricesDf[i-1], 62 | dictofMatricesH1, 63 | i) 64 | 65 | # If feasible, add to the ROA sample collection set. 66 | if (solverSuccessFlag == True): 67 | xFeas = np.vstack((xFeas, xs[:,j].T)) 68 | 69 | if (xFeas.shape[0] == 1): 70 | print("Nothing Feasible!") 71 | else: 72 | # Finally form the approximate ROA. 73 | approxRoa = pc.qhull(xFeas) 74 | 75 | # Plot a non-empty returned approx. ROA. 76 | if (approxRoa.b.shape[0] !=0): 77 | fig = plt.figure() 78 | ax = fig.gca() 79 | approxRoa.plot(ax) 80 | ax.relim() 81 | ax.autoscale_view() 82 | plt.grid(True) 83 | plt.show() 84 | else: 85 | print("CVX hull not full dimensional. Increase Nx.") 86 | -------------------------------------------------------------------------------- /disturbanceFeedbackRmpcLti.py: -------------------------------------------------------------------------------- 1 | # Code implementing the disturbance feedback based robust MPC for LTI systems. 2 | # This code does not do closed-loop MPC. It generates an inner approx. of the ROA. 3 | import numpy as np 4 | import problemDef as pdef 5 | import polytope as pc 6 | from solveMPC import solveMpcDisturbanceFeedback 7 | import matplotlib.pyplot as plt 8 | 9 | # Load the parameters of the problem here. 10 | params = pdef.ProblemParams() 11 | 12 | # Assign the horizon of the MPC. 13 | params.setHorizon(5) 14 | 15 | # Assign the number of initial condition samples to use. 16 | params.setx0SampleCount(576) 17 | 18 | # Compute the terminal set Xn. 19 | params.computeMaxRobustPosInvariantLTI() 20 | 21 | # Compute the big matrices needed. 22 | dictOfMatrices = params.formDfMatrices(params.N, "LTI") 23 | 24 | # Solve MPC from a bunch of sampled initial conditions. 25 | # Store feasible initial conditions as ROA samples. 26 | xs = params.getInitialStateMesh() 27 | xFeas = np.zeros([1, params.nx]) 28 | 29 | for i in range(params.Nx): 30 | solverSuccessFlag = solveMpcDisturbanceFeedback(xs[:,i], 31 | params, 32 | dictOfMatrices, 33 | params.N, 34 | "LTI") 35 | 36 | # If feasible, add to the ROA sample collection set. 37 | if (solverSuccessFlag == True): 38 | xFeas = np.vstack((xFeas, xs[:,i].T)) 39 | 40 | if (xFeas.shape[0] == 1): 41 | print("Nothing Feasible!") 42 | else: 43 | # Finally form the approximate ROA. 44 | approxRoa = pc.qhull(xFeas) 45 | 46 | # Plot a non-empty returned approx. ROA. 47 | if (approxRoa.b.shape[0] !=0): 48 | fig = plt.figure() 49 | ax = fig.gca() 50 | approxRoa.plot(ax) 51 | ax.relim() 52 | ax.autoscale_view() 53 | plt.grid(True) 54 | plt.show() 55 | else: 56 | print("CVX hull not full dimensional. Increase Nx.") 57 | -------------------------------------------------------------------------------- /optimizedTighteningRmpcLpv.py: -------------------------------------------------------------------------------- 1 | # Code implementing the robust MPC with optimization-based constraint tightenings. 2 | # For LPV systems; from the preprint: arxiv.org/abs/2007.00930. 3 | # This code does not do closed-loop MPC. It generates an inner approx. of the ROA. 4 | import numpy as np 5 | from scipy import linalg 6 | import sys 7 | import problemDef as pdef 8 | import polytope as pc 9 | from solveMPC import solveRobustMpcOptimalTightening 10 | import matplotlib.pyplot as plt 11 | 12 | # Load the parameters of the problem here. 13 | params = pdef.ProblemParams() 14 | 15 | # Assign the horizon of the MPC. 16 | params.setHorizon(3) 17 | 18 | # If the horizon is larger than N=4, terminate. 19 | # For this case I suggest using accSimpleRobust.py instead. 20 | if params.N > 4: 21 | print("Lower horizon to N<=4.") 22 | sys.exit() 23 | 24 | # Assign the number of initial condition samples to use. 25 | params.setx0SampleCount(576) 26 | 27 | # Compute the terminal set Xn. 28 | params.computeMaxRobustPosInvariantLPV() 29 | 30 | # Compute the bounds needed. 31 | dictofBounds = params.computeOfflineBounds() 32 | 33 | # The matrices required for the method. 34 | if (params.N > 1): 35 | boldA1Bar = np.zeros([params.nx*params.N, params.nx*params.N]) 36 | for j in range(1,params.N+1): 37 | for k in range(1,j+1): 38 | tmpMat = np.linalg.matrix_power(params.Anom, j-k) if k==1 else \ 39 | np.hstack((tmpMat, np.linalg.matrix_power(params.Anom, j-k))) 40 | 41 | boldA1Bar[params.nx*(j-1): params.nx*j, 0:params.nx*j] = tmpMat 42 | 43 | boldAbar = np.kron(np.eye(params.N), params.Anom) 44 | boldBbar = np.kron(np.eye(params.N), params.Bnom) 45 | Fx = linalg.block_diag(np.kron(np.eye(params.N-1), params.X.A), 46 | params.XnLPV.A) 47 | fx = np.vstack((np.kron(np.ones([params.N-1, 1]), params.X.b), 48 | params.XnLPV.b)) 49 | boldHw = np.kron(np.eye(params.N), params.W.A) 50 | boldhw = np.kron(np.ones([params.N,1]), params.W.b) 51 | boldHu = np.kron(np.eye(params.N), params.U.A) 52 | boldhu = np.kron(np.ones([params.N,1]), params.U.b) 53 | else: 54 | boldAbar = params.Anom 55 | boldBbar = params.Bnom 56 | Fx = params.XnLPV.A 57 | fx = params.XnLPV.b 58 | boldHw = params.W.A 59 | boldhw = params.W.b 60 | boldHu = params.U.A 61 | boldhu = params.U.b 62 | 63 | dictofMatrices = dict(boldA1Bar=boldA1Bar, 64 | boldAbar=boldAbar, 65 | boldBbar=boldBbar, 66 | Fx=Fx, 67 | fx=fx, 68 | boldHw=boldHw, 69 | boldhw=boldhw, 70 | boldHu=boldHu, 71 | boldhu=boldhu) 72 | 73 | # Solve MPC from a bunch of sampled initial conditions. 74 | # Store feasible initial conditions as ROA samples. 75 | xs = params.getInitialStateMesh() 76 | xFeas = np.zeros([1, params.nx]) 77 | 78 | for i in range(params.Nx): 79 | solverSuccessFlag = solveRobustMpcOptimalTightening(xs[:,i], 80 | params, 81 | dictofBounds, 82 | dictofMatrices) 83 | 84 | # If feasible, add to the ROA sample collection set. 85 | if (solverSuccessFlag == True): 86 | xFeas = np.vstack((xFeas, xs[:,i].T)) 87 | 88 | if (xFeas.shape[0] == 1): 89 | print("Nothing Feasible!") 90 | else: 91 | # Finally form the approximate ROA. 92 | approxRoa = pc.qhull(xFeas) 93 | 94 | # Plot a non-empty returned approx. ROA. 95 | if (approxRoa.b.shape[0] !=0): 96 | fig = plt.figure() 97 | ax = fig.gca() 98 | approxRoa.plot(ax) 99 | ax.relim() 100 | ax.autoscale_view() 101 | plt.grid(True) 102 | plt.show() 103 | else: 104 | print("CVX hull not full dimensional. Increase Nx.") 105 | -------------------------------------------------------------------------------- /problemDef.py: -------------------------------------------------------------------------------- 1 | # Define the parameters of the problem and all associated functions here. 2 | import numpy as np 3 | import scipy.signal 4 | from scipy import linalg 5 | from pytope import Polytope 6 | import polytope as pc 7 | import itertools 8 | from controlpy.synthesis import controller_lqr_discrete_time as dlqr 9 | import setOperations as sop 10 | 11 | class ProblemParams: 12 | def __init__(self): 13 | # Dimensions of the state and input spaces. 14 | self.nx = 2 15 | self.nu = 1 16 | 17 | # True dynamics matrices. Used for all LTI examples. 18 | self.A = np.array([[1.0, 0.05], 19 | [0.0, 1.0]]) 20 | self.B = np.array([[0.0], 21 | [1.1]]) 22 | 23 | # Nominal Dynamics matrices. Used for all LPV examples. 24 | self.Anom = np.array([[1.0, 0.15], 25 | [0.1, 1.0]]) 26 | self.Bnom = np.array([[0.1], 27 | [1.1]]) 28 | 29 | # Lists of DeltaA and DeltaB matrices (Draft: arxiv.org/abs/2007.00930). 30 | self.epsA = 0.1 31 | self.epsB = 0.1 32 | 33 | self.delAv = np.array([ [ [0., self.epsA], [self.epsA, 0.] ], 34 | [ [0., self.epsA], [-self.epsA, 0.] ], 35 | [ [0., -self.epsA], [self.epsA, 0.] ], 36 | [ [0., -self.epsA], [-self.epsA, 0.] ] ]) 37 | 38 | self.delBv = np.array([ [ [0.],[-self.epsB] ], 39 | [ [0.],[self.epsB]], 40 | [ [self.epsB], [0.] ], 41 | [ [-self.epsB], [0.]] ]) 42 | 43 | # State constraints. 44 | self.Hx = np.array([[1.0, 0.0], 45 | [-1.0, 0.0], 46 | [0.0, 1.0], 47 | [0.0, -1.0]]) 48 | self.hx = np.array([[8.0], [8.0], [8.0], [8.0]]) 49 | 50 | self.X = Polytope(self.Hx, self.hx) 51 | 52 | # Input constraints. 53 | self.Hu = np.array([[1.0], [-1.0]]) 54 | self.hu = np.array([[4.0], [4.0]]) 55 | 56 | self.U = Polytope(self.Hu, self.hu) 57 | 58 | # Express the constraints as Cx + Du <= b format. 59 | self.C = np.vstack((self.Hx, np.zeros([self.Hu.shape[0], self.nx]))) 60 | self.D = np.vstack( (np.zeros([self.Hx.shape[0], self.nu]), self.Hu) ) 61 | self.b = np.vstack( (self.hx, self.hu) ) 62 | 63 | # Disturbance set. 64 | self.wub = 0.1 65 | self.Hw = np.array([[1.0, 0.0], 66 | [-1.0, 0.0], 67 | [0.0, 1.0], 68 | [0.0, -1.0]]) 69 | self.hw = np.array([[self.wub], [self.wub], [self.wub], [self.wub]]) 70 | 71 | self.W = Polytope(self.Hw, self.hw) 72 | 73 | # Stage cost weight matrices. 74 | self.Q = np.array([[10.0, 0.0], 75 | [0.0, 10.0]]) 76 | self.R = np.array([[2.0]]) 77 | 78 | # MPC horizon, initial condition sample size. Set by user. 79 | self.N = None 80 | self.Nx = None 81 | 82 | # Compute the stabilizing gain K, weight PN (LTI system). 83 | K,self.PN,_ = dlqr(self.A, self.B, 84 | self.Q, self.R) 85 | self.K = -K 86 | 87 | # Compute the stabilizing gain K, weight PN (LPV system). 88 | KvS = scipy.signal.place_poles(self.Anom, 89 | self.Bnom, 90 | np.array([0.745, 0.75])) 91 | self.Kv = -KvS.gain_matrix 92 | 93 | self.PNv = linalg.solve_discrete_lyapunov((self.Anom+\ 94 | self.Bnom@self.Kv).T, self.Q+self.Kv.T@self.R@self.Kv) 95 | 96 | # These invariant sets will be computed when required. 97 | self.E = None 98 | self.XnBar = None 99 | self.XnLTI = None 100 | self.XnLPV = None 101 | 102 | # Assign the horizon of the MPC problem to solve. 103 | def setHorizon(self, N): 104 | self.N = N 105 | 106 | # Assign the number of initial condition samples to use. 107 | def setx0SampleCount(self, Nx): 108 | self.Nx = Nx 109 | 110 | # Generate the initial condition samples via a mesh. 111 | def getInitialStateMesh(self): 112 | x = np.linspace(-9.0, 9.0, int(np.sqrt(self.Nx))) 113 | y = np.linspace(-9.0, 9.0, int(np.sqrt(self.Nx))) 114 | xs = np.empty([self.nx,1]) 115 | 116 | for i in x: 117 | for j in y: 118 | xs = np.hstack((xs, np.array([[i],[j]]))) 119 | 120 | return xs 121 | 122 | # Form the matrices along the horizon needed for disturbance feedback MPC. 123 | # Following the notations of the paper: 124 | # 125 | # Goulart PJ, Kerrigan EC, Maciejowski JM: Optimization over state feedback 126 | # policies for robust control with constraints. Automatica, vol 42, pp 523-33. 127 | # 128 | def formDfMatrices(self, sHorizon, sysFlag): 129 | if (sysFlag == "LTI"): 130 | Xn = self.XnLTI 131 | A = self.A 132 | B = self.B 133 | W = self.W 134 | else: 135 | Xn = self.XnLPV 136 | A = self.Anom 137 | B = self.Bnom 138 | addWBound = self.epsA*self.hx[0].item() + \ 139 | self.epsB*self.hu[0].item() + self.hw[0].item(); 140 | 141 | W = Polytope(lb=-addWBound*np.ones([self.nx,1]), 142 | ub= addWBound*np.ones([self.nx,1])) 143 | 144 | N = sHorizon 145 | dim_t = self.C.shape[0]*N + Xn.A.shape[0] 146 | 147 | boldA = np.eye(self.nx) 148 | for k in range(1,N+1): 149 | boldA = np.vstack( (boldA, np.linalg.matrix_power(A, k) )) 150 | 151 | matE = np.hstack( ( np.eye(self.nx), np.zeros([self.nx, self.nx*(N-1)]) ) ) 152 | boldE = np.vstack( ( np.zeros([self.nx, self.nx*N]), matE ) ) 153 | 154 | for k in range(2,N+1): 155 | matE_updated = np.hstack( (np.linalg.matrix_power(A, k-1), matE[:,0:-self.nx]) ) 156 | boldE = np.vstack( (boldE, matE_updated) ) 157 | matE = matE_updated 158 | 159 | boldB = boldE @ np.kron(np.eye(N), B) 160 | boldC = linalg.block_diag( np.kron( np.eye(N), self.C ), Xn.A ) 161 | szD = np.kron(np.eye(N), self.D).shape[0] 162 | boldD = np.vstack( ( np.kron( np.eye(N), self.D ), np.zeros([dim_t-szD, self.nu*N]) ) ) 163 | 164 | boldF = boldC @ boldB + boldD 165 | boldG = boldC @ boldE 166 | boldH = -boldC @ boldA 167 | smallC = np.vstack( (np.kron(np.ones([N, 1]),self.b), Xn.b ) ) 168 | 169 | # Matrices associated to the stacked disturbances along the horizon. 170 | WAstacked = np.kron(np.eye(N), W.A) 171 | Wbstacked = np.kron(np.ones([N, 1]), W.b) 172 | dim_a = WAstacked.shape[0] 173 | 174 | d = dict(bF=boldF, bG=boldG, bH=boldH, sc=smallC, 175 | wA=WAstacked, wb=Wbstacked, dim_t = dim_t, 176 | dim_a = dim_a) 177 | 178 | return d 179 | 180 | # Compute the minimal RPI set for error (LTI system). 181 | def computeMinRobustPositiveInvariantLTI(self): 182 | maxIter = 100 183 | i = 0 184 | O_v = Polytope(lb = np.zeros((self.nx,1)), ub = np.zeros((self.nx,1))) 185 | 186 | while (i < maxIter): 187 | O_vNext = sop.transformP(self.A + self.B @ self.K, O_v) + self.W; 188 | 189 | # Check if the algorithm has covnerged. 190 | if (i> 0 and O_vNext == O_v): 191 | invariantSet = O_vNext 192 | return invariantSet 193 | 194 | O_v = O_vNext 195 | i = i + 1 196 | 197 | self.E = O_vNext 198 | 199 | # Compute the maximal positive invariant terminal set (Nominal system, LTI). 200 | def computeMaxPosInvariantLTI(self): 201 | # Form the dictionary needed for the precursor function. 202 | dMat = dict(Acl=self.A + self.B @ self.K, 203 | K=self.K, 204 | Hu=self.Hu, 205 | hu=self.hu) 206 | 207 | # Setting a max bound for quitting the iterations. 208 | maxIter = 10 209 | S = self.X - self.E 210 | i = 0 211 | 212 | while (i < maxIter): 213 | pre = sop.preAutLTI(S, dMat) 214 | preIntersectS = pre & S 215 | 216 | if(S == preIntersectS): 217 | return S 218 | else: 219 | S = preIntersectS 220 | i = i + 1 221 | 222 | self.XnBar = S 223 | 224 | # Compute the maximal robust positive invariant terminal set (LTI System). 225 | def computeMaxRobustPosInvariantLTI(self): 226 | # Form the dictionary needed for the precursor function. 227 | dMat = dict(Acl=self.A + self.B @ self.K, 228 | K=self.K, 229 | Wv=self.W.V, 230 | Hu=self.Hu, 231 | hu=self.hu) 232 | 233 | # Setting a max bound for quitting the iterations. 234 | maxIter = 10 235 | S = self.X 236 | i = 0 237 | 238 | while (i < maxIter): 239 | robPre = sop.robustPreAutLTI(S, dMat) 240 | robPreIntersectS = robPre & S 241 | 242 | if(S == robPreIntersectS): 243 | self.XnLTI = S 244 | break 245 | else: 246 | S = robPreIntersectS 247 | i = i + 1 248 | 249 | self.XnLTI = S 250 | 251 | # Compute the maximal robust positive invariant terminal set (LPV System). 252 | def computeMaxRobustPosInvariantLPV(self): 253 | # Form the dictionary needed for the precursor function. 254 | dMat = dict(Anom=self.Anom, 255 | Bnom=self.Bnom, 256 | delAv=self.delAv, 257 | delBv=self.delBv, 258 | K=self.Kv, 259 | Wv=self.W.V, 260 | Hu=self.Hu, 261 | hu=self.hu) 262 | 263 | # Setting a max bound for quitting the iterations. 264 | maxIter = 10 265 | S = self.X 266 | i = 0 267 | 268 | while (i < maxIter): 269 | robPre = sop.robustPreAutLPV(S, dMat) 270 | robPreIntersectS = robPre & S 271 | 272 | if(S == robPreIntersectS): 273 | self.XnLPV = S 274 | break 275 | else: 276 | S = robPreIntersectS 277 | i = i + 1 278 | 279 | self.XnLPV = S 280 | 281 | # Compute the nonconvex offline bounds required for arxiv.org/abs/2007.00930. 282 | def computeOfflineBounds(self): 283 | if self.N == 1: 284 | Fx = self.XnLPV.A 285 | t_w = np.zeros([Fx.shape[0],1]) 286 | t_1 = t_w 287 | t_2 = t_w 288 | t_3 = t_w 289 | t_delTaB = t_w 290 | 291 | # Return all the bounds. 292 | return dict(t_1=t_1, 293 | t_2=t_2, 294 | t_3=t_3, 295 | t_w=t_w, 296 | t_delTaB=t_delTaB) 297 | 298 | # Set of all the possible A matrix vertices. 299 | setA = np.zeros([self.delAv.shape[0], self.nx, self.nx]) 300 | for i in range(self.delAv.shape[0]): 301 | setA[i] = self.Anom + self.delAv[i] 302 | 303 | # Forming the boldA1Bar matrix. 304 | boldA1Bar = np.zeros([self.nx*self.N, self.nx*self.N]) 305 | for j in range(1,self.N+1): 306 | for k in range(1,j+1): 307 | tmpMat = np.linalg.matrix_power(self.Anom, j-k) if k==1 else \ 308 | np.hstack((tmpMat, np.linalg.matrix_power(self.Anom, j-k))) 309 | 310 | boldA1Bar[self.nx*(j-1): self.nx*j, 0:self.nx*j] = tmpMat 311 | 312 | # Forming the boldAvBar matrix. 313 | tmpMat = np.zeros([self.N-1, self.nx*self.N, self.nx*self.N]) 314 | 315 | for n in range(1,self.N): 316 | for j in range(1,self.N+1): 317 | if ((j-1)*self.nx + n*self.nx +1 <= self.nx*self.N): 318 | tmpMat[n-1][(j-1)*self.nx + n*self.nx: j*self.nx + n*self.nx, 319 | (j-1)*self.nx: j*self.nx] = np.eye(self.nx) 320 | 321 | for k in range(tmpMat.shape[0]): 322 | boldAvbar = tmpMat[k] if k==0 else np.hstack((boldAvbar, tmpMat[k])) 323 | 324 | # Form the tdelA and tdelB bounds. 325 | t_dela = float('-inf') 326 | t_delb = float('-inf') 327 | 328 | for j in range(self.delAv.shape[0]): 329 | t_dela = max(t_dela, 330 | np.linalg.norm(np.kron(np.eye(self.N),self.delAv[j]),np.inf)) 331 | 332 | for j in range(self.delBv.shape[0]): 333 | t_delb = max(t_delb, 334 | np.linalg.norm(np.kron(np.eye(self.N),self.delBv[j]),np.inf)) 335 | 336 | # Form the t_delTaB bound. 337 | Fx = linalg.block_diag(np.kron(np.eye(self.N-1), self.X.A), self.XnLPV.A) 338 | t_delTaB = np.zeros([Fx.shape[0], 1]) 339 | for row in range(Fx.shape[0]): 340 | t_delTaB[row] = float('-inf') 341 | for i in range(self.delBv.shape[0]): 342 | t_delTaB[row] = max(t_delTaB[row], t_delb*np.linalg.norm(Fx[row,:]@boldA1Bar@\ 343 | np.kron(np.eye(self.N),self.delBv[i]), 1)) 344 | 345 | # Form all the combinatorial powers of matrices. 346 | APowerMatrices = {int(1):setA} 347 | var = list(range(self.delAv.shape[0])) 348 | for i in range(2,self.N): 349 | lis = [p for p in itertools.product(var, repeat=i)] 350 | APow = np.zeros([len(lis), self.nx, self.nx]) 351 | for j in range(len(lis)): 352 | APow[j] = np.eye(self.nx) 353 | for k in lis[j]: 354 | APow[j] = APow[j] @ setA[k] 355 | 356 | APowerMatrices[int(i)] = APow 357 | 358 | # (N-1)-tuples of indices that are all combinations to pick from APowerMatrices[1-> N-1]. 359 | listOfCombinationsIdx = [] 360 | for i in range(1, self.N): 361 | tmp = APowerMatrices[int(i)] 362 | listOfCombinationsIdx.append(list(range(tmp.shape[0]))) 363 | 364 | combinationMatricesIdx = list(itertools.product(*listOfCombinationsIdx)) 365 | 366 | # Now form the combination (N-1)-tuples of matrices along the horizon using the above ids. 367 | combinationMatrices = np.zeros([len(combinationMatricesIdx), self.nx, (self.N-1)*self.nx]) 368 | for j in range(len(combinationMatricesIdx)): 369 | for k in range(1,self.N): 370 | tmpMat = APowerMatrices[int(k)][combinationMatricesIdx[j][k-1]] if k==1 else \ 371 | np.hstack((tmpMat, 372 | APowerMatrices[int(k)][combinationMatricesIdx[j][k-1]])) 373 | 374 | combinationMatrices[j] = tmpMat 375 | 376 | # Forming all the stacked matrices here for all the above matrix combinations. 377 | delmat = np.zeros([len(combinationMatricesIdx), self.N*self.nx*(self.N-1), self.N*self.nx]) 378 | for j in range(len(combinationMatricesIdx)): 379 | for k in range(1,self.N): 380 | tmpMat = np.kron(np.eye(self.N), combinationMatrices[j][:,(k-1)*self.nx: k*self.nx] \ 381 | -np.linalg.matrix_power(self.Anom,k)) if k==1 else \ 382 | np.vstack((tmpMat, np.kron(np.eye(self.N), 383 | combinationMatrices[j][:,(k-1)*self.nx: k*self.nx] \ 384 | -np.linalg.matrix_power(self.Anom,k)))) 385 | 386 | delmat[j] = tmpMat 387 | 388 | delmat_tw = np.zeros([len(combinationMatricesIdx), self.N*self.nx*(self.N-1), self.N*self.nx]) 389 | for j in range(len(combinationMatricesIdx)): 390 | for k in range(1,self.N): 391 | tmpMat = np.kron(np.eye(self.N), 392 | combinationMatrices[j][:,(k-1)*self.nx: k*self.nx]) if k==1 else \ 393 | np.vstack((tmpMat, np.kron(np.eye(self.N), 394 | combinationMatrices[j][:,(k-1)*self.nx: k*self.nx]))) 395 | delmat_tw[j] = tmpMat 396 | 397 | # Find the bounds by row-wise trying all vertex combinations. 398 | t_0 = np.zeros([Fx.shape[0], 1]) 399 | t_1 = np.zeros([Fx.shape[0], 1]) 400 | t_2 = np.zeros([Fx.shape[0], 1]) 401 | t_3 = np.zeros([Fx.shape[0], 1]) 402 | t_w = np.zeros([Fx.shape[0], 1]) 403 | 404 | for row in range(Fx.shape[0]): 405 | t_w[row] = float('-inf') 406 | t_0[row] = float('-inf') 407 | t_3[row] = float('-inf') 408 | 409 | for j in range(len(combinationMatricesIdx)): 410 | t_w[row] = max(t_w[row], np.linalg.norm(Fx[row,:]@boldAvbar@delmat_tw[j], 1)) 411 | 412 | for j in range(len(combinationMatricesIdx)): 413 | t_0[row] = max(t_0[row], np.linalg.norm(Fx[row,:]@boldAvbar@delmat[j], 1)) 414 | 415 | t_1[row] = t_0[row]*t_dela 416 | t_2[row] = t_0[row]*t_delb 417 | 418 | for j in range(len(combinationMatricesIdx)): 419 | t_3[row] = max(t_3[row], np.linalg.norm(Fx[row,:]@boldAvbar@delmat[j]@ 420 | np.kron(np.eye(self.N), self.Bnom), 1)) 421 | 422 | # Return all the bounds. 423 | return dict(t_1=t_1, 424 | t_2=t_2, 425 | t_3=t_3, 426 | t_w=t_w, 427 | t_delTaB=t_delTaB) 428 | -------------------------------------------------------------------------------- /rigidTubeMpcLti.py: -------------------------------------------------------------------------------- 1 | # Code implementing the rigid tube MPC algorithm. 2 | # This code does not do closed-loop MPC. It generates an inner approx. of the ROA. 3 | import numpy as np 4 | import problemDef as pdef 5 | import polytope as pc 6 | from solveMPC import solveMpcRigidTube 7 | import matplotlib.pyplot as plt 8 | 9 | # Load the parameters of the problem here. 10 | params = pdef.ProblemParams() 11 | 12 | # Assign the horizon of the MPC. 13 | params.setHorizon(5) 14 | 15 | # Assign the number of initial condition samples to use. 16 | params.setx0SampleCount(576) 17 | 18 | # Compute the error invariant E. 19 | params.computeMinRobustPositiveInvariantLTI() 20 | 21 | # Compute terminal set Xn. 22 | params.computeMaxPosInvariantLTI() 23 | 24 | # Solve MPC from a bunch of sampled initial conditions. 25 | # Store feasible initial conditions as ROA samples. 26 | xs = params.getInitialStateMesh() 27 | xFeas = np.zeros([1, params.nx]) 28 | 29 | for i in range(params.Nx): 30 | solverSuccessFlag = solveMpcRigidTube(xs[:,i], params) 31 | 32 | # If feasible, add to the ROA sample collection set. 33 | if (solverSuccessFlag == True): 34 | xFeas = np.vstack((xFeas, xs[:,i].T)) 35 | 36 | if (xFeas.shape[0] == 1): 37 | print("Nothing Feasible!") 38 | else: 39 | # Finally form the approximate ROA. 40 | approxRoa = pc.qhull(xFeas) 41 | 42 | # Plot a non-empty returned approx. ROA. 43 | if (approxRoa.b.shape[0] !=0): 44 | fig = plt.figure() 45 | ax = fig.gca() 46 | approxRoa.plot(ax) 47 | ax.relim() 48 | ax.autoscale_view() 49 | plt.grid(True) 50 | plt.show() 51 | else: 52 | print("CVX hull not full dimensional. Increase Nx.") 53 | -------------------------------------------------------------------------------- /setOperations.py: -------------------------------------------------------------------------------- 1 | # Define all the set operations here. 2 | from pytope import Polytope 3 | import polytope as pc 4 | import numpy as np 5 | import matplotlib.pyplot as plt 6 | 7 | # Function to compute the matrix transformation of a polytope. 8 | def transformP(M, P): 9 | pVertices = P.V 10 | transformedVertices = M@pVertices[0] 11 | 12 | for i in range(pVertices.shape[0]): 13 | ans = np.vstack((transformedVertices, M@pVertices[i])) 14 | transformedVertices = ans 15 | 16 | transformedPolytope = Polytope(transformedVertices) 17 | return transformedPolytope 18 | 19 | # Function to compute the Minkowski sum of two polytopes. 20 | def minkowskiSum(P1, P2): 21 | V_sum = [] 22 | V1 = P1.V 23 | V2 = P2.V 24 | 25 | for i in range(V1.shape[0]): 26 | for j in range(V2.shape[0]): 27 | V_sum.append(V1[i,:] + V2[j,:]) 28 | 29 | return Polytope(np.asarray(V_sum)) 30 | 31 | # Function to compute the Pontryagin difference of two polytopes. 32 | def pontryaginDifference(P1, P2): 33 | Px = P1.A 34 | px = P1.b 35 | p2Vertices = P2.V 36 | hMax = np.zeros([px.size, 1]) 37 | 38 | # Compute the max values row-wise. 39 | for i in range(px.size): 40 | maxVal = float('-inf') 41 | # iterate through P2 vertices. 42 | for j in range(p2Vertices.shape[0]): 43 | maxVal = max(maxVal, Px[i,:] @ p2Vertices[j, :]) 44 | 45 | hMax[i] = maxVal 46 | 47 | pontryaginDifferencePolytope = Polytope(Px, px - hMax) 48 | return pontryaginDifferencePolytope 49 | 50 | # Function to compute the robust precursor set (LTI system). 51 | def robustPreAutLTI(S, dMat): 52 | # Unpack the dictionary. 53 | Acl = dMat["Acl"] 54 | K = dMat["K"] 55 | Wv = dMat["Wv"] 56 | Hu = dMat["Hu"] 57 | hu = dMat["hu"] 58 | 59 | # Initialize. 60 | H = S.A 61 | h = S.b 62 | hTight = np.zeros([h.size, 1]) 63 | 64 | # Compute the tightenings row-wise. 65 | for i in range(h.size): 66 | minTightening = float('inf') 67 | # iterate through W vertices. 68 | for j in range(Wv.shape[0]): 69 | minTightening = min(minTightening, h[i]- H[i,:] @ Wv[j, :]) 70 | 71 | hTight[i] = minTightening 72 | 73 | prePolytope = Polytope(H@Acl, hTight) & Polytope(Hu@K, hu) 74 | 75 | return prePolytope 76 | 77 | # Function to compute the robust precursor set (LPV system). 78 | def robustPreAutLPV(S, dMat): 79 | # Unpack the dictionary. 80 | Anom = dMat["Anom"] 81 | Bnom = dMat["Bnom"] 82 | delAv = dMat["delAv"] 83 | delBv = dMat["delBv"] 84 | K = dMat["K"] 85 | Wv = dMat["Wv"] 86 | Hu = dMat["Hu"] 87 | hu = dMat["hu"] 88 | 89 | # Form all the closed-loop matrices' options. 90 | nx = delAv.shape[1] 91 | clMat = np.zeros([delAv.shape[0]*delBv.shape[0], nx, nx]) 92 | count = 0 93 | for i in range(delAv.shape[0]): 94 | for j in range(delBv.shape[0]): 95 | clMat[count] = (Anom + delAv[i]) + (Bnom+delBv[j]) @ K 96 | count +=1 97 | 98 | # Initialize. 99 | H = S.A 100 | h = S.b 101 | 102 | # Compute the tightenings row-wise for each closed-loop matrix. 103 | for k in range(clMat.shape[0]): 104 | hTight = np.zeros([h.size, 1]) 105 | for i in range(h.size): 106 | minTightening = float('inf') 107 | # iterate through W vertices. 108 | for j in range(Wv.shape[0]): 109 | minTightening = min(minTightening, h[i]- H[i,:] @ Wv[j, :]) 110 | 111 | hTight[i] = minTightening 112 | 113 | prePolytope = Polytope(H@clMat[k], hTight) & Polytope(Hu@K, hu) 114 | 115 | # Do the intersections for all models. 116 | prePolytopeLPV = prePolytope if k==0 else prePolytopeLPV & prePolytope 117 | 118 | 119 | return prePolytopeLPV 120 | 121 | # Function to compute the nominal precursor set (LTI). 122 | def preAutLTI(S, dMat): 123 | # Unpack the dictionary. 124 | Acl = dMat["Acl"] 125 | K = dMat["K"] 126 | Hu = dMat["Hu"] 127 | hu = dMat["hu"] 128 | 129 | # Initialize. 130 | H = S.A 131 | h = S.b 132 | 133 | prePolytope = Polytope(H@Acl, h) & Polytope(Hu@K, hu) 134 | 135 | return prePolytope 136 | -------------------------------------------------------------------------------- /shrinkingTubeMpcLti.py: -------------------------------------------------------------------------------- 1 | # Code implementing the shrinking tube MPC algorithm. 2 | # This code does not do closed-loop MPC. It generates an inner approx. of the ROA. 3 | import numpy as np 4 | import problemDef as pdef 5 | import polytope as pc 6 | from solveMPC import solveMpcShrinkingTube 7 | import matplotlib.pyplot as plt 8 | 9 | # Load the parameters of the problem here. 10 | params = pdef.ProblemParams() 11 | 12 | # Assign the horizon of the MPC. 13 | params.setHorizon(5) 14 | 15 | # Assign the number of initial condition samples to use. 16 | params.setx0SampleCount(576) 17 | 18 | # Compute the terminal set Xn. 19 | params.computeMaxRobustPosInvariantLTI() 20 | 21 | # Solve MPC from a bunch of sampled initial conditions. 22 | # Store feasible initial conditions as ROA samples. 23 | xs = params.getInitialStateMesh() 24 | xFeas = np.zeros([1, params.nx]) 25 | 26 | for i in range(params.Nx): 27 | solverSuccessFlag = solveMpcShrinkingTube(xs[:,i], params) 28 | 29 | # If feasible, add to the ROA sample collection set. 30 | if (solverSuccessFlag == True): 31 | xFeas = np.vstack((xFeas, xs[:,i].T)) 32 | 33 | if (xFeas.shape[0] == 1): 34 | print("Nothing Feasible!") 35 | else: 36 | # Finally form the approximate ROA. 37 | approxRoa = pc.qhull(xFeas) 38 | 39 | # Plot a non-empty returned approx. ROA. 40 | if (approxRoa.b.shape[0] !=0): 41 | fig = plt.figure() 42 | ax = fig.gca() 43 | approxRoa.plot(ax) 44 | ax.relim() 45 | ax.autoscale_view() 46 | plt.grid(True) 47 | plt.show() 48 | else: 49 | print("CVX hull not full dimensional. Increase Nx.") 50 | -------------------------------------------------------------------------------- /solveMPC.py: -------------------------------------------------------------------------------- 1 | # Functions to solve the associated robust MPC problems at any state xt. 2 | from sys import path 3 | # This path to be modified by the user. 4 | path.append(r"/casadi-py27-v3.4.4") 5 | from casadi import * 6 | from scipy import linalg 7 | import setOperations as sop 8 | import matplotlib.pyplot as plt 9 | 10 | # MPC for shrinking tube (LTI system). 11 | def solveMpcShrinkingTube(xt, params): 12 | # Form the closed-loop matrix. 13 | Acl = params.A + params.B @ params.K 14 | 15 | # Optimization variables. 16 | opti = casadi.Opti() 17 | x = opti.variable(params.nx, params.N+1) 18 | u = opti.variable(params.nu, params.N) 19 | 20 | # Initial cost and constraints. 21 | nominalCost = 0. 22 | opti.subject_to( x[:,0] == xt) 23 | 24 | # MPC problem over a horizon of N. 25 | for i in range(params.N): 26 | 27 | # Dynamics of the nominal states. 28 | opti.subject_to( x[:,i+1] == Acl @ x[:,i] + params.B @ u[:,i]) 29 | 30 | # Impose the nominal constraints. 31 | if (i == 0): 32 | nomimalStateConstraintSet = params.X 33 | nomimalInputConstraintSet = params.U 34 | elif (i == 1): 35 | stateConstraintTightening = params.W 36 | inputConstraintTightening = sop.transformP(params.K, params.W) 37 | nomimalStateConstraintSet = params.X - stateConstraintTightening 38 | # Using my own Pontryagin difference function for a 1D input case. 39 | nomimalInputConstraintSet = sop.pontryaginDifference(params.U, 40 | inputConstraintTightening) 41 | else: 42 | stateConstraintTightening = stateConstraintTightening + \ 43 | sop.transformP(np.linalg.matrix_power(Acl,i-1), 44 | params.W) 45 | # Using my own Minkowski sum function for a 1D input case. 46 | inputConstraintTightening = sop.minkowskiSum(inputConstraintTightening, 47 | sop.transformP(params.K @ \ 48 | np.linalg.matrix_power(Acl,i-1), 49 | params.W)) 50 | nomimalStateConstraintSet = params.X - stateConstraintTightening 51 | # Using my own Pontryagin difference function for a 1D input case. 52 | nomimalInputConstraintSet = sop.pontryaginDifference(params.U, 53 | inputConstraintTightening) 54 | 55 | opti.subject_to( nomimalStateConstraintSet.A @ x[:,i] 56 | <= nomimalStateConstraintSet.b ) 57 | opti.subject_to( nomimalInputConstraintSet.A @ (params.K @ x[:,i] + u[:,i]) 58 | <= nomimalInputConstraintSet.b ) 59 | 60 | # Update the cost. 61 | nominalCost += x[:,i].T @ params.Q @ x[:,i] + u[:,i].T @ params.R @ u[:,i] 62 | 63 | # Include the terminal ingredients. 64 | terminalTightening = stateConstraintTightening + \ 65 | sop.transformP(np.linalg.matrix_power(Acl, params.N-1), params.W) 66 | nominalTerminalConstraintSet = params.XnLTI - terminalTightening 67 | 68 | opti.subject_to( nominalTerminalConstraintSet.A @ x[:,params.N] 69 | <= nominalTerminalConstraintSet.b ) 70 | nominalCost += x[:,params.N].T @ params.PN @ x[:,params.N] 71 | 72 | # Solve the MPC problem. 73 | opti.minimize(nominalCost) 74 | opts = {'ipopt.print_level': 0, 'print_time': 0, 'ipopt.sb': 'yes'} 75 | opti.solver('ipopt', opts) 76 | 77 | try: 78 | sol = opti.solve() 79 | except: 80 | return False 81 | 82 | # Return the MPC feasibility flag. 83 | return sol.stats()['return_status'] == 'Solve_Succeeded' 84 | 85 | # MPC for rigid tube (LTI system). 86 | def solveMpcRigidTube(xt, params): 87 | # Form the closed-loop matrix. 88 | Acl = params.A + params.B @ params.K 89 | 90 | # Optimization variables. 91 | opti = casadi.Opti() 92 | x = opti.variable(params.nx, params.N+1) 93 | u = opti.variable(params.nu, params.N) 94 | 95 | # Initial cost and constraint. 96 | nominalCost = 0. 97 | opti.subject_to(params.E.A @ (xt-x[:,0]) <= params.E.b) 98 | 99 | # Form the tightened constraints. 100 | nomimalStateConstraintSet = params.X - params.E 101 | # Using my own Pontryagin difference function for a 1D input case. 102 | nomimalInputConstraintSet = sop.pontryaginDifference(params.U, 103 | sop.transformP(params.K,params.E)) 104 | 105 | # MPC problem over a horizon of N. 106 | for i in range(params.N): 107 | # Dynamics of the nominal states. 108 | opti.subject_to( x[:,i+1] == Acl @ x[:,i] + params.B @ u[:,i]) 109 | 110 | # Impose the nominal constraints. 111 | opti.subject_to( nomimalStateConstraintSet.A @ x[:,i] 112 | <= nomimalStateConstraintSet.b ) 113 | opti.subject_to( nomimalInputConstraintSet.A @ (params.K @ x[:,i] + u[:,i]) 114 | <= nomimalInputConstraintSet.b ) 115 | 116 | # Update the cost. 117 | nominalCost += x[:,i].T @ params.Q @ x[:,i] + u[:,i].T @ params.R @ u[:,i] 118 | 119 | # Include the terminal ingredients. 120 | opti.subject_to( params.XnBar.A @ x[:,params.N] <= params.XnBar.b ) 121 | nominalCost += x[:,params.N].T @ params.PN @ x[:,params.N] 122 | 123 | # Solve the MPC problem. 124 | opti.minimize(nominalCost) 125 | opts = {'ipopt.print_level': 0, 'print_time': 0, 'ipopt.sb': 'yes'} 126 | opti.solver('ipopt', opts) 127 | 128 | try: 129 | sol = opti.solve() 130 | except: 131 | return False 132 | 133 | # Return the MPC feasibility flag. 134 | return sol.stats()['return_status'] == 'Solve_Succeeded' 135 | 136 | # MPC with disturbance feedback policy parametrization. 137 | def solveMpcDisturbanceFeedback(xt, 138 | params, 139 | dictOfMatrices, 140 | sHorizon, 141 | sysFlag): 142 | 143 | boldF = dictOfMatrices["bF"] 144 | boldG = dictOfMatrices["bG"] 145 | boldH = dictOfMatrices["bH"] 146 | smallC = dictOfMatrices["sc"] 147 | WAstacked = dictOfMatrices["wA"] 148 | Wbstacked = dictOfMatrices["wb"] 149 | dim_t = dictOfMatrices["dim_t"] 150 | dim_a = dictOfMatrices["dim_a"] 151 | 152 | # Parse the system and check horizon size. 153 | N = sHorizon 154 | if (sysFlag=="LTI"): 155 | A = params.A 156 | B = params.B 157 | PN = params.PN 158 | else: 159 | A = params.Anom 160 | B = params.Bnom 161 | PN = params.PNv 162 | 163 | # Optimization variables. 164 | opti = casadi.Opti() 165 | 166 | # Policy parametrization matrix. 167 | M = opti.variable(params.nu*N, params.nx*N) 168 | for j in range(params.nu*N): 169 | for k in range(2*j,params.nx*N): 170 | opti.subject_to( M[j,k] == 0.0) 171 | 172 | # Nominal states, inputs, and the dual variables. 173 | x = opti.variable(params.nx, N+1) 174 | v = opti.variable(params.nu*N, 1) 175 | Z = opti.variable(dim_a, dim_t) 176 | 177 | # Initial cost and constraint. 178 | opti.subject_to(x[:,0] == xt.reshape(-1,1)) 179 | opti.subject_to(params.X.A @ x[:,0] <= params.X.b) 180 | nominalCost = 0. 181 | 182 | # Stage cost. 183 | for k in range(N): 184 | x[:,k+1] = A @ x[:, k] + B @ v[k*params.nu:(k+1)*params.nu] 185 | nominalCost += x[:,k].T @ params.Q @ x[:,k] \ 186 | + v[k*params.nu:(k+1)*params.nu].T @ params.R @ v[k*params.nu:(k+1)*params.nu] 187 | 188 | # Include the terminal cost. 189 | nominalCost += x[:,N].T @ PN @ x[:,N] 190 | 191 | # Robust state and input constraints. 192 | opti.subject_to(boldF @ v + Z.T @ Wbstacked <= smallC + boldH @ xt.reshape(-1,1)) 193 | opti.subject_to(vec(Z)>=0.) 194 | opti.subject_to(boldF@M + boldG == Z.T @ WAstacked) 195 | 196 | # Solve the MPC problem. 197 | opti.minimize(nominalCost) 198 | opts = {'ipopt.print_level': 0, 'print_time': 0, 'ipopt.sb': 'yes'} 199 | opti.solver('ipopt', opts) 200 | 201 | try: 202 | sol = opti.solve() 203 | except: 204 | return False 205 | 206 | # Return the MPC feasibility flag. 207 | return sol.stats()['return_status'] == 'Solve_Succeeded' 208 | 209 | # A simple robust MPC for LPV systems. ACC 2021 (Bujarbaruah M., Rosolia R., Sturz Y., Borrelli F.). 210 | def solveMpcAccSimpleRobust(xt, 211 | params, 212 | dictofDFMatrices, 213 | dictofMatricesH1, 214 | sHorizon): 215 | 216 | if (sHorizon > 1): 217 | return solveMpcDisturbanceFeedback(xt, 218 | params, 219 | dictofDFMatrices, 220 | sHorizon, 221 | "LPV") 222 | 223 | # The matrices required. 224 | boldAbar = dictofMatricesH1["boldAbar"] 225 | boldBbar = dictofMatricesH1["boldBbar"] 226 | Fx = dictofMatricesH1["Fx"] 227 | fx = dictofMatricesH1["fx"] 228 | boldHw = dictofMatricesH1["boldHw"] 229 | boldhw = dictofMatricesH1["boldhw"] 230 | boldHu = dictofMatricesH1["boldHu"] 231 | boldhu = dictofMatricesH1["boldhu"] 232 | 233 | # Variables and constraints. 234 | opti = casadi.Opti() 235 | x = opti.variable(params.nx, 2) 236 | v = opti.variable(params.nu,1) 237 | Lambda = opti.variable(Fx.shape[0], boldHw.shape[0]) 238 | 239 | opti.subject_to(x[:,0] == xt) 240 | opti.subject_to(vec(Lambda)>=0.) 241 | opti.subject_to(boldHu@v <= boldhu) 242 | 243 | # Enumerate the set of vertices here. 244 | for i in range(params.delAv.shape[0]): 245 | bolddelA = params.delAv[i] 246 | for j in range(params.delBv.shape[0]): 247 | bolddelB = params.delBv[j] 248 | opti.subject_to(Fx@((boldAbar+bolddelA)@xt + 249 | (boldBbar+bolddelB)@v) + Lambda@boldhw <= fx) 250 | 251 | opti.subject_to(Lambda@boldHw == Fx) 252 | 253 | # Solve the MPC problem. 254 | nominalCost = v.T @ params.R @ v + x[:,1].T @ params.PNv @ x[:,1] 255 | opti.minimize(nominalCost) 256 | opts = {'ipopt.print_level': 0, 'print_time': 0, 'ipopt.sb': 'yes'} 257 | opti.solver('ipopt', opts) 258 | 259 | try: 260 | sol = opti.solve() 261 | except: 262 | return False 263 | 264 | # Return the MPC feasibility flag. 265 | return sol.stats()['return_status'] == 'Solve_Succeeded' 266 | 267 | # Robust MPC for LPV systems with optimization based constraint tightening. 268 | # From draft: arxiv.org/abs/2007.00930. 269 | def solveRobustMpcOptimalTightening(xt, 270 | params, 271 | dictofBounds, 272 | dictofMatrices): 273 | N = params.N 274 | 275 | if (N ==1): 276 | return solveMpcAccSimpleRobust(xt, params, {}, dictofMatrices, N) 277 | 278 | # Bounds computed offline. 279 | t_1 = dictofBounds["t_1"] 280 | t_2 = dictofBounds["t_2"] 281 | t_3 = dictofBounds["t_3"] 282 | t_w = dictofBounds["t_w"] 283 | t_delTaB = dictofBounds["t_delTaB"] 284 | 285 | # The matrices required. 286 | boldA1Bar = dictofMatrices["boldA1Bar"] 287 | boldAbar = dictofMatrices["boldAbar"] 288 | boldBbar = dictofMatrices["boldBbar"] 289 | Fx = dictofMatrices["Fx"] 290 | fx = dictofMatrices["fx"] 291 | boldHw = dictofMatrices["boldHw"] 292 | boldhw = dictofMatrices["boldhw"] 293 | boldHu = dictofMatrices["boldHu"] 294 | boldhu = dictofMatrices["boldhu"] 295 | 296 | # Variables and constraints. 297 | opti = casadi.Opti() 298 | 299 | # Nominal state and inputs. 300 | x = opti.variable(params.nx*(N+1), 1) 301 | v = opti.variable(params.nu*N,1) 302 | 303 | # Policy and dual matrices. 304 | Lambda = opti.variable(Fx.shape[0], boldHw.shape[0]) 305 | M = opti.variable(params.nu*N, params.nx*N) 306 | for j in range(params.nu*N): 307 | for k in range(2*j,params.nx*N): 308 | opti.subject_to(M[j,k] == 0.0) 309 | gamma = opti.variable(boldHw.shape[0], boldHu.shape[0]) 310 | 311 | opti.subject_to(vec(Lambda)>=0.) 312 | 313 | # Input constraints. 314 | opti.subject_to(vec(gamma)>=0.) 315 | opti.subject_to(gamma.T@boldhw <= boldhu - boldHu@v) 316 | opti.subject_to((boldHu@M).T == boldHw.T@gamma) 317 | 318 | # Nominal dynamics evolution. 319 | opti.subject_to(x[0:params.nx] == xt.reshape(-1,1)) 320 | for k in range(1,N+1): 321 | x[k*params.nx:(k+1)*params.nx] = params.Anom@x[(k-1)*params.nx:k*params.nx] +\ 322 | params.Bnom@v[(k-1)*params.nu:k*params.nu] 323 | 324 | # These are done to extract the ininity norms of these opti vectors/matrices. 325 | epsilon_v = opti.variable(1,1) 326 | opti.subject_to(epsilon_v >= 0.) 327 | 328 | # This is for ||v||_inf. 329 | for i in range(v.shape[0]): 330 | # Bounding abs. of each entry of the vector. 331 | opti.subject_to(-epsilon_v <= v[i]) 332 | opti.subject_to(v[i] <= epsilon_v) 333 | 334 | # This is for ||x||_inf. 335 | epsilon_x = opti.variable(1,1) 336 | opti.subject_to(epsilon_x >= 0.) 337 | xNoTermS = x[0:-params.nx] 338 | for i in range(xNoTermS.shape[0]): 339 | # Bounding abs. of each entry of the vector. 340 | opti.subject_to(-epsilon_x <= xNoTermS[i]) 341 | opti.subject_to(xNoTermS[i] <= epsilon_x) 342 | 343 | # This is for ||M||_inf, with matrix M. 344 | epsilon_Mij = opti.variable(params.nu*N, params.nx*N) 345 | epsilon_M = opti.variable(1,1) 346 | opti.subject_to(vec(epsilon_Mij)>= 0.) 347 | opti.subject_to(epsilon_M>=0.) 348 | 349 | # Bounding abs. of each entry of the rows and summing. 350 | rSumVec = opti.variable(M.shape[0], 1) 351 | for i in range(M.shape[0]): 352 | rSum = 0. 353 | for j in range(M.shape[1]): 354 | opti.subject_to(-epsilon_Mij[i,j] <= M[i,j]) 355 | opti.subject_to(M[i,j] <= epsilon_Mij[i,j]) 356 | rSum += epsilon_Mij[i,j] 357 | opti.subject_to(rSumVec[i] == rSum) 358 | 359 | # Bounding the row sum. 360 | opti.subject_to(rSumVec <= epsilon_M*np.ones([M.shape[0], 1])) 361 | 362 | # Verex enumeration here for two terms linear in model mismatches. 363 | for i in range(params.delAv.shape[0]): 364 | bolddelA = np.kron(np.eye(N),params.delAv[i]) 365 | for j in range(params.delBv.shape[0]): 366 | bolddelB = np.kron(np.eye(N),params.delBv[j]) 367 | opti.subject_to(Fx @ boldAbar @ xNoTermS + 368 | Fx @ boldBbar @ v + 369 | Fx @ boldA1Bar @ bolddelA @ xNoTermS + 370 | Fx @ boldA1Bar @ bolddelB @ v + 371 | t_1*epsilon_x + 372 | (t_2+t_delTaB)*(epsilon_M*params.wub) + 373 | t_2*epsilon_v + 374 | t_3*epsilon_M*params.wub + 375 | t_w*params.wub + 376 | Lambda@boldhw <= fx) 377 | 378 | # Dual state constraints. 379 | opti.subject_to(Fx@boldBbar@M + Fx@(boldA1Bar@boldBbar - boldBbar)@M + 380 | Fx@np.eye(boldA1Bar.shape[0]) == Lambda@boldHw) 381 | 382 | # Stage cost and the terminal cost. 383 | nominalCost = x.T @ linalg.block_diag(np.kron(np.eye(N),params.Q), params.PNv) @ x \ 384 | + v.T @ np.kron(np.eye(N), params.R) @ v 385 | 386 | # Solve the MPC problem. 387 | opti.minimize(nominalCost) 388 | opts = {'ipopt.print_level': 0, 'print_time': 0, 'ipopt.sb': 'yes'} 389 | opti.solver('ipopt', opts) 390 | 391 | try: 392 | sol = opti.solve() 393 | except: 394 | return False 395 | 396 | # Return the MPC feasibility flag. 397 | return sol.stats()['return_status'] == 'Solve_Succeeded' 398 | 399 | --------------------------------------------------------------------------------