Transcript MPC
(Real time MPC) Prepared by : Kamal Reza Varhoushi December – 2011
The philosophy behind MPC
Manipulated variables : u(k-1) Controlled variables : y(k) The model of sys used to prediction The critical step of the predictive control
open-loop optimization problem
practical disadvantage of MPC
Robust MPC
The uncertainty representation
Polytopic Uncertainty Structured Feedback Uncertainty
Real time MPC
For each MPC problem, we need to solve an optimization problem for each sampling control law offline ( explicit MPC . So MPC limited to applications with slow dynamics (sampling time in seconds or minutes). A well-known technique for implementing fast MPC is to compute the entire in the form of a lookup table ), in which case the online controller can be implemented as a lookup table. The control action is then implemented online . The major drawback here is that the number of entries in the table can grow exponentially with the horizon, state, and input dimensions, so that “explicit MPC” can only be applied reliably to small problems (where the state dimension is no more than around five). So this method is not applicable for more complex problems .
The new methods for fast MPC, develops by professor Boyd research group in Stanford university. These methods are based on combination of MPC with automatic code generation.
This report describe the capabilities and implementation of CVXGEN software package.
CVXGEN takes a high level description of a convex optimization family, and automatically generate flat, library-free C code that can be compiled into high speed custom solver for the problem family
The main disadvantages of CVXGEN are:
1. It reduced to quadratic programming(QPs) 2. It is suitable for small and medium size problems
Problem statement
energy storage system that can be charged or discharged from a source with varying energy price. A simple example is a battery connected to a power grid The goal is to alternate between charging and discharging in order to maximize the average revenue
Let first introduce problem variables and parameters:
the energy price at time t time horizon T=50 denote discourage excessive charging and discharging
The easiest way to use this interface is via the ‘Matlab’ screen in CVXGEN's online interface, Go to the website “ www.cvxgen.com
” and create new project, then impose your project data’s to the blank spaces shown in window
The second step is creating custom C codes for using in Matlab : Download and extract the ‘cvxgen.zip
’ archive for your problem. This will create a subdirectory called CVXGEN.
Inside the cvxgen/ folder in Matlab, call make_csolve . This will use the mex command to compile and build your custom solver and creates a csolve.mex* file.
As shown in figures, the variation of uc and ud between max and min values, determine this agreement that process has been done in optimal way.
CVXGEN implementation Once the problem is in canonical form, we use a standard primal dual interior point method to find the solution we introduce slack variables to solve equivalent problem
we find analytically the solution of the pair of primal and dual problems: For duality formulation :
Karush-Kuhn-Tucker (KKT) conditions
Each of the primal and dual algorithms require two solves with the so-called KKT matrix to find the solution (L) for the system KL = R.