multi-parametric optimization and control where do we stand? · mp-lp, mp-qp, mp-milp mp-miqp...
TRANSCRIPT
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Multi-parametric Optimization and Control – Where do we stand?
Richard Oberdieck, Nikolaos A. Diangelakis, Ruth Misener, Efstratios N. Pistikopoulos
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Group details and acknowledgment
http://parametric.tamu.edu http://paroc.tamu.edu
We gratefully acknowledge the financial support of EPSRC (EP/M027856/1) Texas A&M University Texas A&M Energy Institute
http://parametric.tamu.edu/http://paroc.tamu.edu/
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Inp
ut
Process‘High Fidelity’ Dynamic Modeling
Output Set-point
Output
Advanced Optimization andControl Policies
Process Modelling to Advanced Optimization and Control Techniques
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Inp
ut
Process‘High Fidelity’ Dynamic Modeling
Output Set-point
Output
Advanced Optimization andControl Policies
No unified platform
No commercially available tool
No generally accepted procedure or ‘protocol’
via
Multi-parametric
Programming
PAROC
Process Modelling to Advanced Optimization and Control Techniques
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Process‘High Fidelity’ Dynamic Modeling
‘High Fidelity’ model Dynamic model
Ordinary Differential Equations
Differential Algebraic Equations
Partial DAE
First Principles Models
High complexity
Often non-linear Custom Models
Advanced Model Libraries
Dynamic and steady-state
simulation
Advanced Optimization
Algorithms
Flowsheeting environment
Process Systems Enterprise, gPROMS, www.psenterprise.com/gproms, 1997-2015
PAROC – PARametric Optimization and ControlA unified framework and software platform
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System Identification
Model Reduction Techniques
Approximate Model
Process‘High Fidelity’ Dynamic Modeling
‘High Fidelity’ model
Model Approximation
Linear state-space models
Model reduction techniques
Statistical methods
Linearization via gPROMS®
Exchange of I/O data via
gO:MATLAB
Execution of gPROMS® model
of arbitrary complexity within
MATLAB®
System Identification Toolbox
PAROC – PARametric Optimization and ControlA unified framework and software platform
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System Identification
Model Reduction Techniques
Approximate Model
Multi-Parametric Programming
Process‘High Fidelity’ Dynamic Modeling
‘High Fidelity’ model
Model Approximation
Multi-Parametric
Programming
Formulation of the optimization
and/or control as a multi-
parametric programming
problem
Explicit map of solutions
mp-LP, mp-QP, mp-MILP
mp-MIQP problems
POP – The Parametric Optimization Toolbox, Pistikopoulos Research Group
http://paroc.tamu.edu/Software
PAROC – PARametric Optimization and ControlA unified framework and software platform
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System Identification
Model Reduction Techniques
Approximate Model
Multi-Parametric Programming
Process‘High Fidelity’ Dynamic Modeling
Multiparametric recedinghorizon policies
‘High Fidelity’ model
Model Approximation
Multi-Parametric
Programming
Multi-Parametric Receding
Horizon Policies
mp-MPC – Control
mp-MHE – State estimation
mp-RHO – Scheduling
PAROC – PARametric Optimization and ControlA unified framework and software platform
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System Identification
Model Reduction Techniques
Approximate Model
Multi-Parametric Programming
Inp
ut
Process‘High Fidelity’ Dynamic Modeling
Output Set-point
Output
Multiparametric recedinghorizon policies
Actions within this area happen once and offline
‘High Fidelity’ model
Model Approximation
Multi-Parametric
Programming
Multi-Parametric Receding
Horizon Policies
Closed-Loop Validation
via gO:MATLAB within
MATLAB®
via C++ within gPROMS®
PAROC – PARametric Optimization and ControlA unified framework and software platform
Focus of this talk
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Multi-parametric Optimization and Control
Nominal
controller
Robust
controller
Continuous
systems
Hybrid systems
What type of system?• Discrete time
• Continuous and hybrid systems• Nominal and robust controllers
?
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Multi-parametric Optimization and Control
Nominal
controller
Robust
controller
Continuous
systems
Hybrid systems
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Multi-parametric Optimization and Control
• 𝑥𝑘+1 = 𝐴𝑥𝑘 + 𝐵𝑢𝑘
• Only continuous variables
Nominal
controller
Robust
controller
Continuous
systems ?
Hybrid systems
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Model Predictive Control (MPC)
Essence: compute the optimal sequence of manipulated variables (inputs) that minimizes
Given: the predicted outputs or states of the system (using a mathematical model) 13
Objective Function = (tracking error, profit, energy etc.)
subject to constraints on inputs and outputs
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Model Predictive Control – how to:
1. At time t, given the measurement y(t) (or state x(t))
2. Solve a Constrained Optimisation Problem to obtain:
a. Predicted future outputs (or states): y(t+1|t), y(t+2|t), … , y(t+P|t)b. Optimal sequence of m.v.: U*={u*(t), u*(t+1), u*(t+2), … , u*(t+m-1)}
3. Apply first input of the sequence u*(t) until time t+1
4. At time t+1 repeat14
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Explicit/multi-parametric MPC
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Treat all uncertainty (initial state, measured disturbance etc.) as parameter
Solve for a range and as a function thereof
Obtain explicit solution of the problem
(2) Critical Regions
(1) Optimal look-up function
mp-QP
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Multi-parametric Programming – An overview
In multi-parametric programming, an optimization problem is solved for a range and as a function of certain parameters
Θ
𝑥 𝜃 = 𝐾𝜃 + 𝑟
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The POP Toolbox – The mp-QP solver
1. Fix 𝜽 = 𝜽𝟎, and solve QPusing the KKT conditions
2. Get parametric solution viaBasic Sensitivity Theorem
3. Define the (critical) regionby optimality and feasibility
4. Cross the facet and findnew 𝜽𝟎
𝐶𝑅0𝐶𝑅1
𝐶𝑅2
𝐶𝑅3
Θ
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Multi-parametric Optimization and Control
Nominal
controller
Robust
controller
Continuous
systemsmp-QP
Hybrid systems
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Multi-parametric Optimization and Control
• 𝑥𝑘+1 = 𝐴𝑥𝑘 + 𝐵𝑢𝑘
• Continuous and discrete variables
Nominal
controller
Robust
controller
Continuous
systemsmp-QP
Hybrid systems ?
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Hybrid Systems – From Model to Optimization
Hybrid systems
Systems with continuous and discrete elements, e.g.
Hybrid systems
Hybrid Model Predictive Controller
The optimal control of hybrid systems results in a MIQP
Hybrid Model Predictive Controller
Multi-parametric MIQP
This problem can be solved explicitly as a mp-MIQP
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mp-MIQP problems – Solution framework
Pre-Processing
Integer Handling
mp-QP solution
Comparison
Termination? STOPNO YES
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mp-MIQP problems – The exact solution
Dua et al. (2002) Axehill et al. (2014)
Oberdieck et al. (2014) Oberdieck and Pistikopoulos (2015)
Comparison over entire CR
Linearization using McCormick
relaxation
The exact solution
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Multi-parametric Optimization and Control
Nominal
controller
Robust
controller
Continuous
systemsmp-QP
Hybrid systemsmp-MIQP
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Multi-parametric Optimization and Control
• 𝑥𝑘+1 = 𝐴(𝑘)𝑥𝑘 + 𝐵(𝑘)𝑢𝑘
• Continuous variables
Nominal
controller
Robust
controller
Continuous
systemsmp-QP ?
Hybrid systemsmp-MIQP
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Robust MPC – Conceptual description
Nominal MPC: Robust MPC:
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Robust MPC – Open-loop versus Closed-loop
Open-loop robust MPC: Closed-loop robust MPC:
• Find a single optimization sequence
that guarantees feasibility over the
entire horizon
• Ignores receding horizon nature of
the problem, i.e. the state is
measured at every stage
Conservative approach
• Identify the states for which
stage-wise feasibility can be
guaranteed
• Reachability analysis, i.e.
Recursive approach, i.e.
dynamic programming is applied
Nominal controller with reduced
feasible space
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Robust MPC – The uncertainty set Ω
General polytope Box-constrained
Extreme points only via verticesExtreme points available in
halfspace representation
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Robust Counterpart – Key concept
Robust counterpart
Reformulation of a robust optimization problem into anequivalent (regular) optimization problem
Robust counterpart
Key simplification
Instead of general polytopic Ω, we consider
Key simplification
The reformulation
This allows us to write the following:
Ben-Tal and Nemirovski (2000) Robust solutions of Linear Programming problems contaminated with uncertain data.
Mathematical Programming 88(3), 411 – 424.
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Robust mp-MPC – Example problem
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Multi-parametric Optimization and Control
Nominal
controller
Robust
controller
Continuous
systemsmp-QP mp-LPs + mp-QP
Progress
Hybrid systemsmp-MIQP
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Multi-parametric Optimization and Control
Nominal
controller
Robust
controller
Continuous
systemsmp-QP mp-LPs + mp-QP
Progress
Hybrid systemsmp-MIQP ?
• 𝑥𝑘+1 = 𝐴(𝑘)𝑥𝑘 + 𝐵(𝑘)𝑢𝑘
• Continuous and discrete variables
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Robust hybrid mp-MPC – Conceptual developments
Apply the same principle: but what changed? The variable space 𝒱 becomes discontinuous (and thus non-
convex):𝒱 = ℝ𝑛 × {0,1}𝑝
Thus, the stage-wise problem becomes mp-MILP This means the reachability analysis becomes more complicated:
…
𝒳𝑁 = 𝒳𝒳𝑁−1 =
𝑖∈ℐ
𝒳𝑁−1𝑖
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Multi-parametric Optimization and Control
Nominal
controller
Robust
controller
Continuous
systemsmp-QP mp-LPs + mp-QP
Progress
Hybrid systemsmp-MIQP mp-MILPs + mp-MIQPs
Progress
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Multi-parametric Optimization and control – Conclusion
We presented1. An overview over the state-of-the-art in multi-parametric
optimization and control2. Recent results on the exact solution of mp-MIQP problems3. An intuitive way to solve closed-loop robust mp-MPC problems4. The extension to robust hybrid mp-MPC
In future1. Develop automated implementation of robust hybrid mp-MPC
code2. Validate in similar fashion to robust mp-MPC approach3. Tighten the suboptimality of the robust counterpart using novel
counterpart descriptions
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Multi-parametric Optimization and Control – Where do we stand?
Richard Oberdieck, Nikolaos A. Diangelakis, Ruth Misener, Efstratios N. Pistikopoulos