The model predictive controller QP solver converts a linear MPC optimization problem to the general form QP problem
$$\underset{x}{Min}(\frac{1}{2}{x}^{\u22ba}Hx+{f}^{\u22ba}x)$$
subject to the linear inequality constraints
$$Ax\le b$$
where
x is the solution vector.
H is the Hessian matrix. This matrix is constant when your prediction model and tuning weights do not change at run time.
A is a matrix of linear constraint coefficients. This matrix is constant when your prediction model does not change at run time.
b and f are vectors.
At the beginning of each control interval, the controller computes H, f, A, and b. If H or A is constant, the controller retrieves their precomputed values.
Model Predictive Control Toolbox™ software supports two builtin algorithms for solving the QP problem. Both solvers require the Hessian matrix to be positive definite.
Activeset solver — This solver can provide fast and robust performance for
smallscale and mediumscale optimization problems in both single and double precision.
The activeset solver uses the KWIK algorithm from [1]. To use the activeset solver,
set the Optimizer.Algorithm
property of your MPC controller to
'activeset'
. To configure the algorithm settings, use the
Optimizer.ActiveSetOptions
property of your controller.
Interiorpoint solver — This solver can provide superior performance for largescale
optimization problems, such as MPC applications that enforce constraints over large
prediction and control horizons. This interiorpoint solver uses a primaldual algorithm
with a Mehrotra predictorcorrector. To use the interiorpoint solver, set the
Optimizer.Algorithm
property of your MPC controller to
'interiorpoint'
. To configure the algorithm settings, use the
Optimizer.InteriorPointOptions
property of your controller.
When selecting and configuring the QP solver for your application, consider the following:
The size and configuration of the MPC problem affects the performance of the builtin QP solvers. To determine which solver is best for your application, consider simulating your controller across multiple simulation scenarios using both QP solvers.
The interiorpoint solver is more sensitive to solver parameters than the activeset solver. Therefore, it can require more adjustment to find an optimal balance between performance and robustness.
The activeset solver also uses a nonadjustable tolerance when testing for an optimal solution. You can adjust the optimality tolerances for the interiorpoint solver.
One or more linear constraints can be violated slightly due to numerical roundoff
errors. Such violations are normal and do not generate warning messages. To adjust the
tolerance for acceptable constraint violations, use the
ConstraintTolerance
setting for either the activeset or
interiorpoint solver.
The search for a QP solution is an iterative process. For either solver, you can
specify the maximum number of iterations using the corresponding
MaxIterations
setting. If the number of iterations reaches the
maximum, the algorithm terminates.
The default maximum number of iterations for the activeset solver is $$4\left({n}_{c}+{n}_{v}\right)$$, where n_{c} and
n_{v} are the number of constraints and
optimization variables across the prediction horizon, respectively. For some
controller configurations, this value can be very large, which can make the QP solver
appear to stop responding. This value has a lower bound of
120
.
The default maximum number of iterations for the interiorpoint solver is
50
.
If your MPC problem includes hard constraints after conversion to a QP problem, the QP inequality constraints can be infeasible (impossible to satisfy). If the QP solver detects infeasibility, it terminates immediately.
When the solver detects an infeasible problem or reaches the maximum number of
iterations without finding an optimal solution, the controller retains the last successful
control output. For more information, see mpcmove
. You can detect an abnormal outcome and override the default
behavior as you see fit.
In the first control step, the QP solvers use a cold start, in which the initial guess is the unconstrained solution described in Unconstrained Model Predictive Control. If x satisfies the constraints, it is the optimal QP solution and the algorithm terminates. Otherwise, at least one of the linear inequality constraints must be satisfied as an equality, and the solver computes the optimal solution. For subsequent control steps:
The activeset solver uses a warm start where the active constraint set determined in the previous control step becomes the initial guess.
The interiorpoint solver continues to use a cold start.
For a given MPC application with constraints, there is no way to predict how many QP solver iterations are required to find an optimal solution. Also, in realtime applications, the number of iterations can change dramatically from one control interval to the next. In such cases, the worstcase execution time can exceed the limit that is allowed on the hardware platform and determined by controller sample time.
You set a guaranteed worstcase execution time for your MPC controller by applying a
suboptimal solution after the number of optimization iterations exceeds a specified
maximum value. To set the worstcase execution time, first determine the time needed for a
single optimization iteration by experimenting with your controller under nominal
conditions. Then, set an upper bound on the number of iterations per control interval. For
example, if it takes around 1 ms to compute each iteration on the hardware and the
controller sample time is 10 ms, set the maximum number of iterations to no greater than
10
.
MPCobj.Optimizer.ActiveSetOptions.MaxIterations = 10;
By default, an MPC controller object has a lower bound of 120
on
the maximum number of iterations for the activeset solver.
By default, when the solver reaches the maximum number of solver iterations without an
optimal solution, the controller holds the manipulated variables at their previous values.
To use the suboptimal solution reached after the final iteration, set the
UseSuboptimalSolution
option to true
.
MPCobj.Optimizer.UseSuboptimalSolution = true;
While the solution is not optimal, the MPC controller adjusts the solution such that it satisfies all your specified constraints.
There is no guarantee that the suboptimal solution performs better than if you hold the controller output constant. You can simulate your system using both approaches, and select the configuration that provides better controller performance.
For an example, see Use Suboptimal Solution in Fast MPC Applications.
To access the QP solvers for applications that require solving online QP problems, use
the mpcActiveSetSolver
and mpcInteriorPointSolver
functions, which are useful for:
Advanced MPC applications that are beyond the scope of Model Predictive Control Toolbox software.
Custom QP applications, including applications that require code generation.
Model Predictive Control Toolbox software lets you specify a custom QP solver for your MPC controller. This solver is called in place of the builtin solvers at each control interval. This option is useful for:
Validating your simulation results or generating code with an inhouse thirdparty solver that you trust.
Applications where the builtin solvers do not provide satisfactory performance for your specific problem.
You can define a custom solver for simulation or for code generation. In either instance, you define the custom solver using a custom function and configure your controller to use this custom function.
Task  Custom Solver Function  Affected MATLAB^{®} Functions  Affected Simulink^{®} Blocks 

Simulation Set

Supports:



Code Generation Set

Supports:


To simulate an MPC controller with a custom QP solver, perform the following steps.
Copy the solver template file to your working folder or anywhere on the
MATLAB path, and rename it mpcCustomSolver.m
. To copy the
solver template to your current working folder, type the following at the MATLAB command line.
src = which('mpcCustomSolver.txt'); dest = fullfile(pwd,'mpcCustomSolver.m'); copyfile(src,dest,'f');
Modify mpcCustomSolver.m
by adding your own custom solver. Your
solver must be able to run in MATLAB and be implemented in a MATLAB script or MEX file.
Configure your MPC controller MPCobj
to use the custom
solver.
MPCobj.Optimizer.CustomSolver = true;
The software now uses your custom solver for simulation in place of the builtin QP KWIK solver.
Simulate your controller. For more information, see Simulation.
For an example, see Simulate MPC Controller with a Custom QP Solver.
You can generate code for MPC controllers that use a custom QP solver written in either C/C++ code or MATLAB code suitable for code generation.
To do so at the command line, you must have MATLAB Coder™ software.
To do so in Simulink you must have Simulink Coder or Simulink PLC Coder™ software.
To generate code for MPC controllers that use a custom QP solver, perform the following steps.
Copy the solver template file to your working folder or anywhere on the
MATLAB path, and rename it mpcCustomSolverCodeGen.m
. To copy
the MATLAB code template to your current working folder, type the following at the
MATLAB command line.
src = which('mpcCustomSolverCodeGen_TemplateEML.txt'); dest = fullfile(pwd,'mpcCustomSolverCodeGen.m'); copyfile(src,dest,'f');
Alternatively, you can use the C template.
src = which('mpcCustomSolverCodeGen_TemplateC.txt'); dest = fullfile(pwd,'mpcCustomSolverCodeGen.m'); copyfile(src,dest,'f');
Modify mpcCustomSolverCodeGen.m
by adding your own custom
solver.
Configure your MPC controller MPCobj
to use the custom
solver.
MPCobj.Optimizer.CustomSolverCodeGen = true;
The software now uses your custom solver for code generation in place of the builtin QP KWIK solver.
Generate code for the controller. For more information, see Generate Code and Deploy Controller to RealTime Targets.
For an example, see Simulate and Generate Code for MPC Controller with Custom QP Solver.
You can implement the same custom QP solver for both simulation and code generation. To do so:
Set both Optimizer.CustomSolver
and
Optimizer.CustomSolverCodeGen
to true
.
Create both mpcCustomSolver.m
and
mpcCustomSolverCodeGen.m
.
During simulation, your controller uses the mpcCustomSolver.m
custom function. For code generation, your controller uses the
mpcCustomSolverCodeGen.m
custom function.
You can specify the same MATLAB code in both custom solver functions, provided the code is suitable for code generation.
If you implement mpcCustomSolverCodeGen.m
using C/C++ code, create
a MEX file using the code. You can then call this MEX file from
mpcCustomSolver.m
. For more information on creating and using MEX
files, see C MEX File Applications.
When you implement a custom QP solver, your custom function must have one of the following signatures:
Custom solver for simulation:
function [x,status] = mpcCustomSolver(H,f,A,b,x0)
Custom solver for code generation:
function [x,status] = mpcCustomSolverCodeGen(H,f,A,b,x0)
For both simulation and code generation, your custom solver has the following input and output arguments.
H
is a Hessian matrix, specified as an
nbyn symmetric positive definite matrix, where
n is the number of optimization variables.
f
is the multiplier of the objective function linear term,
specified as a column vector of length n.
A
is a matrix of linear inequality constraint coefficients,
specified as an mbyn matrix, where
m is the number of constraints.
b
is the right side of the inequality constraint equation,
specified as a column vector of length m.
x0
is an initial guess for the solution, specified as a column
vector of length n.
x
is the optimal solution, returned as a column vector of
length n.
status
is a solution validity indicator, returned as an integer
as shown in the following table.
Value  Description 

> 0  x is optimal. status represents
the number of iterations performed during optimization. 
0  The maximum number of iterations was reached without finding an
optimal solution. The solution If the

1  The problem appears to be infeasible, that is, the constraints cannot be satisfied. 
2  An unrecoverable numerical error occurred. 
Note
The MPC controller expects the custom solver functions to solve the QP problem
subject to the linear inequality constraints $$Ax\ge b$$. If your custom solver uses $$Ax\le b$$, you must change the sign of both A
and
b
before passing them to your custom solver code.
You can use FORCES PRO, a realtime embedded optimization software tool developed by Embotech AG, to simulate and generate code for a linear MPC controller designed using Model Predictive Control Toolbox software. Starting in FORCES PRO 2.0, Embotech provides a plugin that leverages the design capabilities of Model Predictive Control Toolbox software and the computational performance of FORCES PRO. Using the plugin, you can generate a custom QP solver that allows deployment on realtime hardware and is highly optimized based on your specific MPC problem to achieve satisfactory realtime performance. Especially longhorizon MPC problems can be solved very efficiently.
For information on the FORCES PRO solver, visit the Embotech website. For information on using the FORCES PRO solver together with Model Predictive Control Toolbox software, see the FORCES PRO documentation.
[1] Schmid, C., and L.T. Biegler. "Quadratic Programming Methods for Reduced Hessian SQP." Computers & Chemical Engineering 18, no. 9 (September 1994): 817–32. https://doi.org/10.1016/00981354(94)E00014.
mpc
 mpcActiveSetSolver
 mpcInteriorPointSolver
 mpcmove