Configure Optimization Solver for Nonlinear MPC
By default, nonlinear MPC controllers solve a nonlinear programming problem using the
fmincon
function with the SQP algorithm, which requires Optimization Toolbox™ software. Alternatively, you can specify a continuation/generalized minimum
residual (C/GMRES) solver or your own custom nonlinear solver, neither of which require
Optimization Toolbox software.
Solver Decision Variables
For nonlinear MPC controllers at time t_{k}, the nonlinear optimization problem uses the following decision variables:
Predicted state values from time t_{k+1} to t_{k+p}. These values correspond to rows 2 through p+1 of the
X
input argument of your cost and constraint functions, where p is the prediction horizon.Predicted manipulated variables from time t_{k} to t_{k+p1}. These values correspond to the manipulated variable columns in rows 1 through p of the
U
input argument of your cost and constraint functions.
Therefore, the number of decision variables N_{Z} is equal to p(N_{x}+N_{mv}) + 1, N_{x} is the number of states, N_{mv} is the number of manipulated variables, and the +1 accounts for the global slack variable.
The problem formulation used here, in which all the intermediate state variables are
part of the vector of decision variables, is called a sparse
formulation. It is also known as a simultaneous or
multipleshooting formulation. In this formulation, equality
constraints of the type x(k+1) =
f(x(k),u(k))
must be enforced. This formulation results in larger problems with many equality
constraints, but it leads to a sparse, regular, structure. Such
structure yields better numerical stability and conditioning for large horizons and for
unstable or nonsmooth plants. It can also result in a much more efficient optimization when
you use a (possibly parallel) solver that can exploit sparsity. The sparse formulation is
the default used by Model Predictive Control Toolbox™ for nonlinear MPC problems (when fmincon
is used as a
solver).
The sparse formulation differs from the dense formulation, in which equality constraints such as x(k+1) = f(x(k),u(k)) are solved using numerical simulation, and the intermediate state variables are eliminated by substitution into the cost and constraint functions, thereby leaving just the manipulated variables (plus a slack variable) as decision variables of the optimization problem.
The dense formulation is also referred to as a sequential or singleshooting formulation, and is used by Model Predictive Control Toolbox for the C/GMRES solver. A dense formulation is also the default for linear MPC problems, as described in Optimization Problem.
Specify Initial Guesses
A properly configured standard linear MPC optimization problem has a unique solution. However, nonlinear MPC optimization problems often allow multiple solutions (local minima), and finding a solution can be difficult for the solver. In such cases, it is important to provide a good starting point near the global optimum.
During closedloop simulations, it is best practice to warm start your nonlinear solver. To do so, use the predicted state and manipulated variable trajectories from the previous control interval as the initial guesses for the current control interval. In Simulink^{®}, the Nonlinear MPC Controller block is configured to use these trajectories as initial guesses by default. To use these trajectories as initial guesses at the command line:
Return the
opt
output argument when callingnlmpcmove
. Thisnlmpcmoveopt
object contains any runtime options you specified in the previous call tonlmpcmove
. It also includes the initial guesses for the state (opt.X0
) and manipulated variable (opt.MV0
) trajectories, and the global slack variable (opt.Slack0
).Pass this object in as the
options
input argument tonlmpcmove
for the next control interval.
These commandline simulation steps are best practices, even if you do not specify any other runtime options.
Configure fmincon
Options
By default, nonlinear MPC controllers optimize their control move using the
fmincon
function from the Optimization Toolbox. When you first create your controller, the
Optimization.SolverOptions
property of the nlmpc
object contains the standard fmincon
options with the following
nondefault settings:
Use the
SQP
algorithm (SolverOptions.Algorithm = 'sqp'
)Use objective function gradients (
SolverOptions.SpecifyObjectiveGradient = 'true'
)Use constraint gradients (
SolverOptions.SpecifyConstraintGradient = 'true'
)Do not display optimization messages to the command window (
SolverOptions.Display = 'none'
)
These nondefault options typically improve the performance of the nonlinear MPC controller.
You can modify the solver options for your application. For example, to specify the
maximum number of solver iterations for your application, set
SolverOptions.MaxIter
. For more information on the available solver
options, see fmincon
(Optimization Toolbox).
In general, you should not modify the SpecifyObjectiveGradient
and
SpecifyConstraintGradient
solver options, since doing so can
significantly affect controller performance. For example, the constraint gradient matrices
are sparse, and setting SpecifyConstraintGradient
to false would cause
the solver to calculate gradients that are known to be zero.
Configure C/GMRES Solver Options
The C/GMRES method is another builtin option for optimization. The computational efficiency and convergence properties of the C/GMRES method are particularly useful when applied to large scale systems and systems that exhibit nonlinear and nonconvex constraints.
To use the C/GMRES method to solve multistage nonlinear MPC problems, specify the
Optimization.Solver
property of your multistage nonlinear MPC object
as "cgmres"
. When you do so, the software stores a default C/GMRES
options object in the Optimization.SolverOptions
property of your
multistage MPC object.
The C/GMRES options object has the following properties, which you can modify using dot notation.
BarrierParameter
— Parameter barrier value. This is a positive scalar representing the strength of the penalty term added to the constraints in the barrier function method for nonlinear model predictive control. The penalty term is used to penalize lower and upped bounds constraint violations. The default value is0.1
.Display
— Level of display. This is a string or character vector specified as either"iter"
or"none"
. Use"iter"
to display solver information at every iteration, and use"none"
to suppress any solver display. The default value is"none"
.FiniteDifferenceStepSize
— Step size used for finite differences. This is a positive scalar representing the step size that the forward finite difference scheme uses to approximate the Jacobian in the optimization problem. A larger step size can result in faster convergence but may also lead to numerical instability. The default value is1e8
.MaxIterations
— Maximum number of iterations allowed. This is a positive integer representing the maximum number of innerloop iterations allowed for the solver. The default value is100
.Restart
— Number of outerloop iterations. This is a nonnegative integer representing the maximum number of outerloop iterations for the solver. The default value is10
.StabilizationParameter
— Stabilization parameter. This is a positive scalar representing a forcing term that you can use to control the step size and improve the numerical stability of the solver.In inexact Newton methods, the Newton direction is typically approximated using an iterative linear solver, such as the conjugate gradient method or the GMRES method. However, due to computational limitations or other factors, the iterative solver may not find an exact solution to the linear system within a specified tolerance. The forcing term is introduced to compensate for the inexactness of the Newton direction. It adjusts the step size taken in each iteration to ensure convergence, even when the Newton direction is not accurate enough. The forcing term can be viewed as a trust region radius that limits the step size based on the accuracy of the Newton direction. Therefore a decrease of the stabilization parameter normally leads to a decrease of the step size.
If the
StabilizationParameter
is too large, it may cause overcorrection and oscillations, leading to slow convergence or even divergence. On the other hand, if theStabilizationParameter
is too small, it may result in slow convergence as the step size is overly conservative. To enhance algorithm convergence, it is advisable to select aStabilizationParameter
value of 1/Ts, where Ts represents the model sampling time. This parameter should not exceed 2/Ts.The default value is
1e3
.TerminationTolerance
— Termination tolerance. This is a positive scalar representing the termination tolerance for the solver. The solver stops when the relative error between two consecutive iterations is less than the termination tolerance. The default value is1e6
.
For an example on how to use the C/GMRES solver, see Control Robot Manipulator Using C/GMRES Solver and Solve FuelOptimal Navigation Problem using C/GMRES.
Specify Custom Solver
As an alternative to the fmincon
function and C/GMRES method, you
can specify your own custom nonlinear solver. To do so, create a custom wrapper function
that converts the interface of your solver function to match the interface expected by the
nonlinear MPC controller. Your custom function must be a MATLAB^{®} script or MATfile on the MATLAB path. For an example that shows a template custom solver wrapper function, see
Optimizing Tuberculosis Treatment Using Nonlinear MPC with a Custom Solver.
You can use the Nonlinear Programming solver developed by Embotech AG to simulate and generate code for nonlinear MPC controllers. For more information, see Implement MPC Controllers Using Embotech FORCESPRO Solvers.
To configure your nlmpc
object to use your custom solver wrapper
function, set its Optimization.CustomSolverFcn
property in one of the
following ways:
Name of a function in the current working folder or on the MATLAB path, specified as a string or character vector
Optimization.CustomSolverFcn = "myNLPSolver";
Handle to a function in the current working folder or on the MATLAB path
Optimization.CustomSolverFcn = @myNLPSolver;
Your custom solver wrapper function must have the signature:
function [zopt,cost,flag] = myNLPSolver(FUN,z0,A,B,Aeq,Beq,LB,UB,NLCON)
This table describes the inputs and outputs of this function, where:
N_{Z} is the number of decision variables.
M_{cineq} is the number of linear inequality constraints.
M_{ceq} is the number of linear equality constraints.
N_{cineq} is the number of nonlinear inequality constraints.
N_{ceq} is the number of nonlinear equality constraints.
Argument  Input/Output  Description 

FUN  Input  Nonlinear cost function to minimize, specified as a handle to a function with the signature: [F,G] = FUN(z) and arguments:

z0  Input  Initial guesses for decision variable values, specified as a vector of length N_{Z} 
A  Input  Linear inequality constraint array, specified as an
M_{cineq}byN_{Z}
array. Together, A and B define constraints of
the form $$\text{A}\cdot z\le \text{B}$$. 
B  Input  Linear inequality constraint vector, specified as a column vector of length
M_{cineq}. Together, A
and B define constraints of the form $$\text{A}\cdot z\le \text{B}$$. 
Aeq  Input  Linear equality constraint array, specified as an
M_{ceq}byN_{Z}
array. Together, Aeq and Beq define
constraints of the form $$\text{Aeq}\cdot z=\text{Beq}$$. 
Beq  Input  Linear equality constraint vector, specified as a column vector of length
M_{ceq}. Together, Aeq
and Beq define constraints of the form $$\text{Aeq}\cdot z=\text{Beq}$$. 
LB  Input  Lower bounds for decision variables, specified as a column vector of length N_{Z}, where $$\text{LB}\left(i\right)\le z\left(i\right)$$. 
UB  Input  Upper bounds for decision variables, specified as a column vector of length N_{Z}, where $$z\left(i\right)\le \text{UB}\left(i\right)$$. 
NLCON  Input  Nonlinear constraint function, specified as a handle to a function with the signature: [cineq,c,Gineq,Geq] = NLCON(z) and arguments:

zopt  Output  Optimal decision variable values, returned as a vector of length N_{Z}. 
cost  Output  Optimal cost, returned as a scalar. 
flag  Output  Exit flag, returned as one of the following:

When you implement your custom solver function, it is best practice to have your solver use the cost and constraint gradient information provided by the nonlinear MPC controller.
If you are unable to obtain a solution using your custom solver, try to identify a special condition for which you know the solution, and start the solver at this condition. If the solver diverges from this initial guess:
Check the validity of the state and output functions in your prediction model.
If you are using a custom cost function, make sure it is correct.
If you are using the standard MPC cost function, verify the controller tuning weights.
Make sure that all constraints are feasible at the initial guess.
If you are providing custom Jacobian functions, validate your Jacobians using
validateFcns
.
See Also
Functions
fmincon
(Optimization Toolbox) validateFcns