Multi-parametric Optimization and
Control – Where do we stand?
Richard Oberdieck, Nikolaos A. Diangelakis, Ruth
Misener, Efstratios N. Pistikopoulos
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
Process Modelling to Advanced Optimization and Control
Techniques
Output
Input
Process
‘High Fidelity’ Dynamic Modeling
Advanced Optimization and
Control Policies
Output Set-point
Process Modelling to Advanced Optimization and Control
Techniques
Process
‘High Fidelity’ Dynamic Modeling
Output
 No unified platform
 No commercially
available tool
 No generally accepted
procedure or
‘protocol’
Input
PAROC
via
Multi-parametric
Programming
Advanced Optimization and
Control Policies
Output Set-point
PAROC – PARametric Optimization and Control
A unified framework and software platform
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 Control
A unified framework and software platform
Process
‘High Fidelity’ Dynamic Modeling


‘High Fidelity’ model
Model Approximation
System
Identification

Linear state-space models
 Model reduction techniques
 Statistical methods
 Linearization via gPROMS®
Model Reduction
Techniques
Approximate Model



Exchange of I/O data via
gO:MATLAB
Execution of gPROMS® model
of arbitrary complexity within
MATLAB®
System Identification Toolbox
PAROC – PARametric Optimization and Control
A unified framework and software platform
Process
‘High Fidelity’ Dynamic Modeling



‘High Fidelity’ model
Model Approximation
Multi-Parametric
Programming

Formulation of the optimization
and/or control as a multiparametric programming
problem
 Explicit map of solutions
 mp-LP, mp-QP, mp-MILP
mp-MIQP problems
System
Identification
Model Reduction
Techniques
Approximate Model
Multi-Parametric Programming
POP – The Parametric Optimization Toolbox, Pistikopoulos Research Group
http://paroc.tamu.edu/Software
PAROC – PARametric Optimization and Control
A unified framework and software platform
Process
‘High Fidelity’ Dynamic Modeling




‘High Fidelity’ model
Model Approximation
Multi-Parametric
Programming
Multi-Parametric Receding
Horizon Policies
mp-MPC – Control
 mp-MHE – State estimation
 mp-RHO – Scheduling
System
Identification
Model Reduction
Techniques
Approximate Model

Multi-Parametric Programming
Multiparametric receding
horizon policies
PAROC – PARametric Optimization and Control
A unified framework and software platform
Process
‘High Fidelity’ Dynamic Modeling




‘High Fidelity’ model
Model Approximation
Multi-Parametric
Programming
Multi-Parametric Receding
Horizon Policies
Closed-Loop Validation
System
Identification
Input


via gO:MATLAB within
MATLAB®
 via C++ within gPROMS®
Focus of this talk
Output
Model Reduction
Techniques
Approximate Model
Multi-Parametric Programming
Multiparametric receding
horizon policies
Output Set-point
Actions within this area happen once and offline
Multi-parametric Optimization and Control
What type of system?
• Discrete time
• Continuous and hybrid systems
• Nominal and robust controllers
Nominal
controller
Continuous
systems
Hybrid systems
?
Robust
controller
Multi-parametric Optimization and Control
Nominal
controller
Continuous
systems
Hybrid systems
Robust
controller
Multi-parametric Optimization and Control
Nominal
controller
Continuous
systems
?
Hybrid systems
• 𝑥𝑘+1 = 𝐴𝑥𝑘 + 𝐵𝑢𝑘
• Only continuous variables
Robust
controller
Model Predictive Control (MPC)
 Essence: compute the optimal sequence of manipulated variables (inputs) that
minimizes
Objective Function = (tracking error, profit, energy etc.)
subject to constraints on inputs and outputs
 Given: the predicted outputs or states of the system (using a mathematical
13
model)
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.
b.
Predicted future outputs (or states): y(t+1|t), y(t+2|t), … , y(t+P|t)
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 repeat
14
Explicit/multi-parametric MPC
 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
(1) Optimal look-up function
(2) Critical Regions
mp-QP
15
Multi-parametric Programming – An overview
In multi-parametric programming, an optimization
problem is solved for a range and as a function of
certain parameters
𝑥 𝜃 = 𝐾𝜃 + 𝑟
Θ
The POP Toolbox – The mp-QP solver
1. Fix 𝜽 = 𝜽𝟎 , and solve QP
using the KKT conditions
2. Get parametric solution via
Basic Sensitivity Theorem
3. Define the (critical) region
by optimality and feasibility
4. Cross the facet and find
new 𝜽𝟎
𝐶𝑅2
𝐶𝑅1
𝐶𝑅0
𝐶𝑅3
Θ
Multi-parametric Optimization and Control
Nominal
controller
Continuous
systems
Hybrid systems
mp-QP
Robust
controller
Multi-parametric Optimization and Control
Nominal
controller
Continuous
systems
Hybrid systems
Robust
controller
mp-QP
?
• 𝑥𝑘+1 = 𝐴𝑥𝑘 + 𝐵𝑢𝑘
• Continuous and discrete variables
Hybrid Systems – From Model to Optimization
Hybrid systems
Systems
with continuous
and discrete elements, e.g.
Hybrid Model
Predictive Controller
The
optimal control
of hybrid systems results in a MIQP
Multi-parametric
MIQP
This problem can be solved explicitly as a mp-MIQP
mp-MIQP problems – Solution framework
Pre-Processing
Integer Handling
mp-QP solution
Comparison
NO
Termination?
YES
STOP
mp-MIQP problems – The exact solution
Dua et al. (2002)
Axehill et al. (2014)
Comparison over entire CR
Linearization using McCormick
relaxation
Oberdieck et al. (2014)
Oberdieck and Pistikopoulos (2015)
The exact solution
Multi-parametric Optimization and Control
Nominal
controller
Continuous
systems
Hybrid systems
mp-QP
mp-MIQP
Robust
controller
Multi-parametric Optimization and Control
Continuous
systems
Hybrid systems
Nominal
controller
Robust
controller
mp-QP
?
mp-MIQP
• 𝑥𝑘+1 = 𝐴(𝑘)𝑥𝑘 + 𝐵(𝑘)𝑢𝑘
• Continuous variables
Robust MPC – Conceptual description
Nominal MPC:
Robust MPC:
Robust MPC – Open-loop versus Closed-loop
Open-loop robust MPC:
• Find a single optimization sequence
that guarantees feasibility over the
entire horizon
Closed-loop robust MPC:
• Identify the states for which
stage-wise feasibility can be
guaranteed
• Reachability analysis, i.e.
Recursive approach, i.e.
dynamic programming is applied
• Ignores receding horizon nature of
the problem, i.e. the state is
measured at every stage
Conservative approach
Nominal controller with reduced
feasible space
Robust MPC – The uncertainty set Ω
General polytope
Extreme points only via vertices
Box-constrained
Extreme points available in
halfspace representation
Robust Counterpart – Key concept
Robust counterpart
Reformulation
of a robust optimization problem into an
Key
Key simplification
simplification
equivalent (regular) optimization problem
Instead of general polytopic Ω, we consider
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.
Robust mp-MPC – Example problem
Multi-parametric Optimization and Control
Continuous
systems
Hybrid systems
Nominal
controller
Robust
controller
mp-QP
mp-LPs + mp-QP
Progress
mp-MIQP
Multi-parametric Optimization and Control
Continuous
systems
Hybrid systems
Nominal
controller
Robust
controller
mp-QP
mp-LPs + mp-QP
Progress
mp-MIQP
• 𝑥𝑘+1 = 𝐴(𝑘)𝑥𝑘 + 𝐵(𝑘)𝑢𝑘
• Continuous and discrete variables
?
Robust hybrid mp-MPC – Conceptual developments
 Apply the same principle: but what changed?
 The variable space 𝒱 becomes discontinuous (and thus nonconvex):
𝒱 = ℝ𝑛 × {0,1}𝑝
 Thus, the stage-wise problem becomes mp-MILP
 This means the reachability analysis becomes more complicated:
…
𝑖
𝒳𝑁−1
𝒳𝑁−1 =
𝑖∈ℐ
𝒳𝑁 = 𝒳
Multi-parametric Optimization and Control
Continuous
systems
Hybrid systems
Nominal
controller
Robust
controller
mp-QP
mp-LPs + mp-QP
Progress
mp-MIQP
mp-MILPs + mp-MIQPs
Progress
Multi-parametric Optimization and control – Conclusion
 We presented
1. An overview over the state-of-the-art in multi-parametric
optimization and control
2. Recent results on the exact solution of mp-MIQP problems
3. An intuitive way to solve closed-loop robust mp-MPC problems
4. The extension to robust hybrid mp-MPC

In future
1. Develop automated implementation of robust hybrid mp-MPC
code
2. Validate in similar fashion to robust mp-MPC approach
3. Tighten the suboptimality of the robust counterpart using novel
counterpart descriptions
Multi-parametric Optimization and
Control – Where do we stand?
Richard Oberdieck, Nikolaos A. Diangelakis, Ruth
Misener, Efstratios N. Pistikopoulos