MPT 2.6.2 Changelog

Included latest version of YALMIP

The latest release of YALMIP is now included in the distribution of MPT 2.6.2. Highlights of the new version:

Added the function binmodel which converts polynomial expressions with binary variables to linear expressions by introducing additional variables and constraints.
Added max-plus control example.
Improved performance on most nonconvex quadratic programs in bmibnb.
Fixed various bugs in the robust optimization framework.

Included latest version of the Ellipsoidal Toolbox

The Ellipsoidal Toolbox version 1.1 is now included in the distribution of MPT. Highlights of the new version:

Additional operations with ellipsoids
Maxmin and minmax reach sets for discrete-time systems with disturbances
Fixed numerous bugs
Updated manual

Massive speedup of the mpQP algorithm

The mpQP algorithm provided by MPT has been significantly improved and now it runs up to 2-10 times faster compared to the version contained in MPT 2.6.1. As shown in the table below, the speedup grows with an increasing dimension of the parametric dimension.

Parameter dimension	Runtime MPT 2.6.1	Runtime MPT 2.6.2	Improvement
`4D (1700 regions)`	242 secs	71 secs	3.4x
`8D (1000 regions)`	124 secs	63 secs	2x
`18D (10 regions)`	44 secs	4 secs	11x

Notice that the numbers have been obtained for concrete examples, by no means they should be interpreted such that, in general, it is easier to solve an 18D example than an 8D setup.

Mex version of `polytope/isinside`

MPT 2.6.2 now includes a C-coded version of the polytope/isinside method, which significantly faster compared to its MATLAB-coded counterpart. We remind the users that the function isinside(Pn, x0) searches a given polytope array and returns true if the array Pn contains the given point x0. Therefore if you are interested in increasing the speed of evaluation of explicit control laws in the MATLAB environment, searching through polytopes, or decreasing runtimes of parametric solvers (mpLP, mpQP), we strongly recommend that you compile the provided C-code by running the following code on your MATLAB prompt:

>> cd mpt/@polytope
>> mex isinside.c
>> cd ..

The table below illustrates the efficiency of the new implementation.

Task	Runtime Matlab version	Runtime Mex version	Improvement
`isinside(Pn,x0)` where `Pn` has 20000 regions	0.45 secs	0.01 secs	45x
`sim(ctrl,x0,100)` where `ctrl` has 1700 regions	3.2 secs	0.4 secs	8x
mpQP algorithm, nx=4, nu=20	173 secs	160 secs	8%

New function `mpt_mpsol2ctrl`

YALMIP users are now able to convert parametric solutions obtained by the solvemp() function into MPT's controller objects for further post-processing. The conversion is achieved by calling the function mpt_mpsol2ctrl. There are two options:

>> sol = solvemp(F, obj, options, x, u)
>> ctrl = mpt_mpsol2ctrl(sol, nu)

where nu is the number of system inputs, or

>> sol = solvemp(F, obj, options, x, u)
>> ctrl = mpt_mpsol2ctrl(sol, sysStruct, probStruct)

where sysStruct and probStruct are MPT's system and problem structures, respectively. Consult the output of help mpt_mpsol2ctrl for more details.

New function `polytope/grid`

With the help of a new function grid(P, npoints) you will be able to place several equidistantly placed points into a given polytope/ polytope array P. The points are distributed in a way such that each axis is first divided into a total of npoints points, and only the points which are contained in P are returned. For instance:

>> P = unitbox(2);
>> X = grid(P, 3)

X =

    -1    -1
    -1     0
    -1     1
     0    -1
     0     0
     0     1
     1    -1
     1     0
     1     1

Improved referencing of polytope arrays

Polytope arrays can now be indexed using logicals, e.g.

>> P = [unitbox(1) 2*unitbox(1)]
>> Q = P(logical([0 1]))
>> Q == P(2)

ans =

     1

You are now able to obtain multiple copies of the same polytope by repeating the same index several times, e.g.

>> P = unitbox(2);
>> Q = P([1 1 1])
Q=

Polytope array: 3 polytopes in 2D

Needless to say, MPT 2.6.2 significantly improved the speed of any polytope indexing, on average by a factor of 2.

Faster solver for mixed-integer problems

The latest version of the MIQP solver in YALMIP (which is used as a fall-back solver if you don't have CPLEX or XPRESS installed) uses a new heuristic to find feasible solutions during the branching process. On average, solutions are obtained 2x faster when applied to typical MPC problems. It should however be noted that the performance of the solver depends on a concrete example. If you see a significant slowdown, you can always switch to the original version. To do that, first open the file |mpt_init.m| and go to line 438 which reads:

bnb_upper_solver = 'fixer';

and change it to

bnb_upper_solver = 'rounder';

Then run mpt_init from the MATLAB command line to re-initialize the settings.

Notable improvements and bug fixes

MATLAB R2006b compatibility fixes
Improve qhull numerical robustness in matlab R14 and newer
Speed improvements of the mpMILP/mpMIQP algorithms
mpt_mplp: properly handle the F*x term in the cost
mpt_mplp: improved detection of infeasible cases
mpt_mpqp: improved search for an initial feasible point
mpt_mpqp: automatically symmetrize the Hessian
mpt_ownmpc: Open-loop solution had wrong number of elements
mpt_simplify: properly reject/accept controllers with overlaps
mpt_simplify: greedy merging failed for 1D polytopes
mpt_simplify: make optimal merging more numerically robust
mpt_sys: fixed an undefined variable
mpt_verify: speed improvements
isinvariant/isstabilizable was broken in MPT 2.6.1
polytope/isinside: auto-switch to fastbreak=1
polytope/subsref: speedup
polytope/uminus didn't update bounding box
tracking=1 failed for on-line MPC for MLD systems
Simulink interface: improved detection of tracking controllers
Suppress annoying warnings when using quadprog