1. Distributed Stochastic Model Predictive Control

2. Outline 1. Background 2. DSMPC ── Additive Uncertainty
3. DSMPC ── Parameter Uncertainty
4. DSMPC ── Additive Uncertainty & Parameter Uncertainty

3. Intelligent Transportation Intelligence Agriculture
1. Background
Internet of Vehicles
Chemical Process
Smart Grid
Intelligent Transportation
Intelligence Agriculture
Intelligent Community

4. Distributed Stochastic
1. Background
System Characteristics：
Multiple Subsystems
Wide Distribution
Multiple Constraints
Various Uncertainties
The control problem
of a group of subsystems subject to uncertainties and coupled constraints
Distributed Stochastic
MPC

5. 1. Background
云控制背景
Xia Yuanqing. From networked control systems to cloud control systems [C]. In: Proceedings of the 31st Chinese Control Conference (CCC), Hefei, China,
2012:
Xia Yuanqing. Cloud control systems [J]. IEEE/CAA Journal of Automatica Sinica, 2015, 2(2):
夏元清. 云控制系统及其面临的挑战[J]. 自动化学报, 2016，42(1):1-12.
Xia Yuanqing, Qin Y, Zhai D H, et al. Further results on cloud control systems[J]. Science China Information Sciences, 2016, 59(7): 1-5.

6. Robust V.S. Stochastic MPC
1. Background
Robust V.S. Stochastic MPC
Disturbances
Measurement Error
Model Uncertainty
Robust MPC：
Worst-case uncertainties
Hard constraints
Stochastic MPC：
Some known statistical description of the uncertainty
(probability distribution, or the first and second moments)
Probabilistic constraints

7. Centralized V.S. Distributed MPC
1. Background
Centralized V.S. Distributed MPC
Centralized MPC:
The complete system is modeled, and all the control inputs are computed in one optimization problem.
Computational complexity
Communication bandwidth limitations
Reliance on a single processor
Distributed MPC:
Local control decisions are computed using local measurements and reduced-order models of the local dynamics.
The dynamic performance of it is worse than centralized framework

8. Distributed Stochastic MPC
1. Background
Distributed Stochastic MPC
The research of DSMPC is still in an embryonic stage, and lots of difficult but rather important problems still remain to be solved.
How to coordinate efforts to ensure that the distributed decisions lead to coupled constraint satisfaction.
How to utilize the probabilistic information to design controllers which ensure recursive feasibility and closed-loop stability.
How to utilize the relationship between the global objective and the independent decision-making of subsystems to achieve a coordinated response of the entire system.

9. Outline 1. Background 2. DSMPC ── Additive Uncertainty
3. DSMPC ── Parameter Uncertainty
4. DSMPC ── Additive Uncertainty & Parameter Uncertainty

10. 2.1. Problem statement
Remark
Consider a system of Np linear uncertain subsystems:
assumed i.i.d. for each subsystem p
: zero-mean, independent, with known distributions
Orthotope,

11. 2.1. Problem statement Local probabilistic constraints:
Nc coupling constraints across multiple subsystems:
Open loop prediction strategy:

12. 2.1. Problem statement State decomposition: : nominal : uncertain
Remark
belong to a tube centred on :

13. 2.2. Centralized SMPC 【Cost function】 The system-wide cost function:
Remark
The system-wide cost function:
where
Finite cost
minimize numerically online
Quadratic in :
computed offline

14. 2.2. Centralized SMPC 【Constraint handling strategy】
A first step towards guaranteeing that the constraints are met in closed-loop operation is to ensure that the constraints are satisfied by the predicted state and input sequences for each subsystem at all time k.

15. 2.2. Centralized SMPC Theorem 1
Remark
Tightened linear constraints on nominal input/state predictions
Given the distribution of
= sum of independent r.v.'s

16. 2.2. Centralized SMPC
Although the conditions of Theorem 1 ensure that constraints are satisfied over the entire prediction horizon for each subsystem p and each constraint c∈ C at time k, the existence of cp(k) satisfying constraints does not ensure the existence of cp(k+1) generating predictions at time k+1 satisfying constraints. Consequently Theorem 1 does not guarantee the future feasibility of an online optimization problem incorporating the conditions in Theorem 1 as constraints.
Consider the i-step-ahead prediction at time k:
at time k+1, this term has already been realized
where

17. 2.2. Centralized SMPC Consider the i-step-ahead prediction at time k:
worst case bound
probabilistic bound
where

18. 2.2. Centralized SMPC Theorem 2 (a)
Predictions at time k must ensure feasibility at k+1, k+2, . . .
Hence tighten local constraints on nominal i-step-ahead prediction by ,where = maximum element of ith column of: k
k+1
k+2
Remark
Satisfaction of local probabilistic constraints and recursive feasibility is ensured if
Theorem 2 (a)

19. 2.2. Centralized SMPC Theorem 2 (b)
Tighten coupling constraints on nominal i-step-ahead prediction by ,where = maximum element of ith column of:
where k
k+1
k+2
Remark
Satisfaction of coupling probabilistic constraints and recursive feasibility is ensured if
Theorem 2 (b)

20. 2.2. Centralized SMPC
【CMPC optimization】
Remark

21. 2.2. Centralized SMPC
Largest element of each column lies on the diagonal:

22. 2.2. Centralized SMPC

23. 2.3. Distributed SMPC 【Distributed strategy】
Centralized
Distributed
Remark
With the whole system at a state x(k), only one subsystem p is permitted to update at this time step.
the subsystem optimizing at time k If
the new plan is obtained as the solution to the local optimization problem.
Otherwise

24. 2.3. Distributed SMPC 【Terminal constraint】
Constraints in MPC optimization at time k for
can be computed using e.g. [Gilbert & Tan, 1991]
is convex

25. 2.3. Distributed SMPC 【DMPC optimization】
Remark
At a time step k, the local optimization problem for pk :
Quadratic Program (QP)

26. 2.3. Distributed SMPC
【Main results】
Remark
Theorem 3

27. 2.3. Distributed SMPC
Proof:
Theorem 2

28. 2.3. Distributed SMPC Define the global cost as a Lyapunov function
Summing over r time-steps:
Hence

29. 2.4. Numerical example Model parameters:
Four probabilistic constraints including local constraints and state coupling constraints:
Disturbance derived from truncated normal distributions:

30. 2.4. Numerical example The LQ-optimal gain: Prediction parameters:
Prediction horizon:
Terminal sets:
Simulation parameters:
1000 realizations of disturbance sequences
Initial conditions:

31. 2.4. Numerical example Local constraints and coupled constraints:
Unconstrained
optimal control
DSMPC

32. 2.5. Conclusion Additive Uncertainty
Coupling probabilistic constraints are handled in a distributed way.
The property of recursive feasibility is guaranteed with respect to both local and coupling probabilistic constraints.
The stability of the large-scale system is analyzed in the presence of additive stochastic disturbances.
Li Dai, Yuanqing Xia, Yulong Gao, Mark Cannon. Distributed stochastic MPC of linear systems with additive uncertainty and coupled probabilistic. IEEE Transactions on Automatic Control, 2017, 62(7),

33. Outline 1. Background 2. DSMPC ── Additive Uncertainty
3. DSMPC ── Parameter Uncertainty
4. DSMPC ── Additive Uncertainty & Parameter Uncertainty

34. 3.1. Theoretical background on gPCEs
Generalized polynomial chaos expansions (gPCEs) provide means to uniformly approximate any second-moment processes, which apply to most physical processes.
n i.i.d. random variables
Remark
the highest order’s of the polynomials

35. 3.1. Theoretical background on gPCEs
Once the coefficients are computed, by applying the orthogonality property of multivariate polynomials, the desired moments of can be computed directly from the coefficients of its gPCE.

36. 3.2. Problem statement
Consider a system of Np stochastic discrete-time linear subsystems:
Remark

37. 3.2. Problem statement Local probabilistic constraints:
Nc coupling constraints across multiple subsystems:
Local and coupled hard constraints on the inputs:

38. 3.3. DSMPC algorithm with gPCEs
The local problem for subsystem p:
Remark
Remark
To render the online solution computationally feasible, in the following, gPCEs are used to present a tractable formulation for the optimization problem.

39. 3.3. DSMPC algorithm with gPCEs
First consider the approximation of stochastic dynamics using gPCEs methods.

40. 3.3. DSMPC algorithm with gPCEs
Remark
Galerkin projection
Deterministic linear differential equations:
Remark

41. 3.3. DSMPC algorithm with gPCEs
We next consider the cost function in the gPCEs framework.
The control input as follows:

42. 3.3. DSMPC algorithm with gPCEs
Remark
RemarkC
The cost function can be written as a quadratic form of
Remark:

43. 3.3. DSMPC algorithm with gPCEs
A more challenging difficulty that arises in practice is the need to approximate probabilistic constraints.
Remark

44. 3.3. DSMPC algorithm with gPCEs
Remark
Proof.
Lemma 4.1 44

45. 3.3. DSMPC algorithm with gPCEs
Remark
We consider the presence of probabilistic constraints on both states and inputs of the following form:
Convex second-order cone constraints:

46. order cone constraints
3.3. DSMPC algorithm with gPCEs
Remark
convex second-
order cone constraints
convex sets
quadratic function
Remark
Problem 2 is a convex optimization problem that can be solved very efficiently.

47. 3.4. Properties of gPCEs-based DSMPC
invariant set

48. 3.4. Properties of gPCEs-based DSMPC
The next result shows that the resulting closed-loop system has the properties of probabilistic constraint satisfaction and recursive feasibility.

49. 3.4. Properties of gPCEs-based DSMPC
The stability of such stochastic systems can be determined by examining the stability of the moments of the solution, which are deterministic functions.

50. 3.5. Numerical example Model parameters:
Four probabilistic constraints including local constraints and state coupling constraints:
Initial conditions:

51. 3.5. Numerical example Simulation parameters: Weighting metrices:
Terminal sets:
100 realizations of disturbance sequences

52. 3.5. Numerical example Input constraints：
local input constraint boundary
Unconstrained optimal control
gPCEs-based DSMPC

53. 3.5. Numerical example Coupled probabilistic constraints：
coupled constraint boundary
Unconstrained optimal control
gPCEs-based DSMPC
Remark
gPCEs-based DSMPC: k = 0: 0% k = 1: 18% k = 2: 17%
k = 3: 16% k = 4: 18%

54. Parameter Uncertainty
3.6. Conclusion
Parameter Uncertainty
The proposed approximation results in a deterministic convex optimization for the control policy.
The proposed gPCEs-based DSMPC algorithm guarantees both local and coupled probabilistic constraints satisfaction at each step.
The property of recursive feasibility is guaranteed with respect to constraints.
The algorithm ensures stochastic stability of the overall closed-loop system irrespective of how the model uncertainty influences the system’s matrices.
Li Dai, Yuanqing Xia, Yulong Gao. Distributed MPC of linear systems with stochastic parametric uncertainties and coupled probabilistic constraints. SIAM Journal on Control and Optimization, 2015, 53(6):

55. Outline 1. Background 2. DSMPC ── Additive Uncertainty
3. DSMPC ── Parameter Uncertainty
4. DSMPC ── Additive Uncertainty & Parameter Uncertainty

56. 4.1. Problem statement
Remark
Consider a system of Np dynamically decoupled subsystems
is i.i.d. for each subsystem p
are zero-mean, independent, with known distributions:

57. 4.1. Problem statement Predicted control input sequence:
Local probabilistic constraints:
Nc coupling constraints across multiple subsystems:

58. 4.2. DSMPC Algorithm 【Update strategy】
Only one subsystem p is permitted to update at each time step.
the optimizing subsystem at time k If
the new plan is obtained as the solution to the local optimization problem.
Otherwise, adopt the feasible perturbation sequences

59. 4.2. DSMPC Algorithm 【Constraint handling strategy】
To construct an implementable optimization problem, a strategy to handle the probabilistic constraints is first investigated.
【Constraint handling strategy】
Remark
State decomposition:
: nominal
: uncertain
with

60. 4.2. DSMPC Algorithm a linear time-invariant system with disturbance
Remark
a linear time-invariant system with disturbance

61. 4.2. DSMPC Algorithm K
Since the evolution of is described by a linear time-invariant system with disturbance, the conditions which ensure the satisfaction of probabilistic constraints are given below.
Remark

62. 4.2. DSMPC Algorithm
Remark

63. 4.2. DSMPC Algorithm
Remark

64. 4.2. DSMPC Algorithm K
The dynamics of ζp involves both the nominal states and the perturbations. To handle ζp, a sequence of sets (i.e. a tube) with bounding facets of fixed orientation is constructed online to bound ζp.
Remark
Suppose that the tube has polytopic cross-sections and is parameterized by

65. 4.2. DSMPC Algorithm
Remark

66. 4.2. DSMPC Algorithm
Remark

67. 4.2. DSMPC Algorithm 【Construction of terminal sets】
To formulate the MPC problem with a finite number of constraints, terminal sets are constructed for

68. 4.2. DSMPC Algorithm
Remark 1. 2. 3.

69. 4.2. DSMPC Algorithm
Remark
Remark
Terminal constraints:

70. 4.2. DSMPC Algorithm 【Optimization problem formulation】
Remark1
Define the cost function in terms of the nominal state as
Remark
a quadratic function of the optimization variables

71. 4.2. DSMPC Algorithm
Remark
MPC optimization problem can now be formulated as follows.
Remark
The optimization problem is a convex optimization problem, which can be solved very efficiently.

72. 4.2. DSMPC Algorithm
【Main results】
Remark

73. 4.3. Numerical example Model parameters: s
Disturbance is truncated from a normal distribution : s
Four probabilistic constraints including local constraints and state coupling constraints:

74. 4.3. Numerical example Weights: Prediction parameters:
Prediction horizon:
Prediction parameters:
Simulation parameters:
1000 realizations of the uncertainty sequences
Initial conditions:
Weights:
Update sequence: f
Tube parameters:

75. 4.3. Numerical example Subsystem 1：
DSMPC
Unconstrained optimal control
Under the unconstrained optimal control, the violation probability of local probabilistic constraint is 100% at time k = 1. While under DSMPC algorithm, 5.1% the same uncertainty realizations violate the local constraint.

76. 4.3. Numerical example Subsystem 2：
DSMPC
Unconstrained optimal control
Under the unconstrained optimal control, the violation probability of local probabilistic constraint is 100% at time k = 1. While under DSMPC algorithm, 1% the same uncertainty realizations violate the local constraint.

77. 4.3. Numerical example Subsystem 3：
DSMPC
Unconstrained optimal control
Under the unconstrained optimal control, the violation probability of local probabilistic constraint is 100% at time k = 1. While under DSMPC algorithm, 2.5% the same uncertainty realizations violate the local constraint.

78. 4.3. Numerical example Coupled constraint：
DSMPC
Unconstrained optimal control
Under the unconstrained optimal control, the violation probability of coupled probabilistic constraint is 100% at time k = 1. While under DSMPC algorithm, the coupled constraint is violated with a probability of 5.3%.

79. Parameter Uncertainty
4.4. Conclusion
Additive Uncertainty
Parameter Uncertainty
Coupling probabilistic constraints are handled in a distributed way.
The formulated MPC optimization problem for each subsystem is a convex optimization.
The property of recursive feasibility is guaranteed with respect to both local and coupling probabilistic constraints.
The stability of the large-scale system is analyzed in the presence of parameter uncertainty and disturbances.
Li Dai, Yuanqing Xia, Yulong Gao, Mark Cannon. Distributed stochastic MPC for systems with parameter uncertainty and disturbances. International Journal of Robust and Nonlinear Control; 2018, 28(6),

80. References
Li Dai, Yuanqing Xia, Yulong Gao, Mark Cannon. Distributed stochastic MPC of linear systems with additive uncertainty and coupled probabilistic. IEEE Transactions on Automatic Control, 2017, 62(7),
Li Dai, Yuanqing Xia, Yulong Gao, Basil Kouvaritakis, Mark Cannon. Cooperative distributed stochastic MPC of linear systems with state estimation and coupled probabilistic constraints. Automatica, 2015, 61:
Li Dai, Yuanqing Xia, Yulong Gao. Distributed MPC of linear systems with stochastic parametric uncertainties and coupled probabilistic constraints. SIAM Journal on Control and Optimization, 2015, 53(6):
Li Dai, Yuanqing Xia, Yulong Gao, Mark Cannon. Distributed stochastic MPC for systems with parameter uncertainty and disturbances. International Journal of Robust and Nonlinear Control, 2018, 28(6),
Li Dai, Yuanqing Xia, Yulong Gao, Mark Cannon. Distributed stochastic MPC of linear systems with parameter uncertainty and disturbances. Proceedings of the 35th Chinese Control Conference, 2016,

81. Acknowledgements
Li Dai was born in Beijing, China, in She received the B.S. degree in Information and Computing Science in 2010 and the Ph.D. degree in Control Science and Engineering in 2016 from Beijing Institute of Technology, Beijing, China. Now she is an assistant professor in the School of Automation of Beijing Institute of Technology. Her research interests include model predictive control, distributed control, data-driven control, stochastic systems, and networked control systems.
Yulong Gao was born in Henan Province, China, in He received the B.S. degree in Automation in 2013 and the M.S. degree in Control Science and Engineering in 2016 from Beijing Institute of Technology. He is pursuing the Ph.D. degree in KTH Royal Institute of Technology. His research interests include networked control system, model predictive control, nonlinear system, and multi-agent system.

82. Acknowledgements
Collaborators:
Mark Cannon
Basil Kouvaritakis

83. Thanks for your attention！