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 tk, the nonlinear optimization problem uses the following decision variables:
Predicted state values from time tk+1 to tk+p. These values correspond to rows 2 through p+1 of the
Xinput argument of your cost and constraint functions, where p is the prediction horizon.
Predicted manipulated variables from time tk to tk+p-1. These values correspond to the manipulated variable columns in rows 1 through p of the
Uinput argument of your cost and constraint functions.
Therefore, the number of decision variables NZ is equal to p(Nx+Nmv) + 1, Nx is the number of states, Nmv is the number of manipulated variables, and the +1 accounts for the global slack variable.
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 closed-loop 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:
optoutput argument when calling
nlmpcmoveoptobject contains any run-time options you specified in the previous call to
nlmpcmove. It also includes the initial guesses for the state (
opt.X0) and manipulated variable (
opt.MV0) trajectories, and the global slack variable (
Pass this object in as the
optionsinput argument to
nlmpcmovefor the next control interval.
These command-line simulation steps are best practices, even if you do not specify any other run-time 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
object contains the standard
fmincon options with the following
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
fmincon (Optimization Toolbox).
In general, you should not modify the
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 built-in 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 non-convex constraints.
To use the C/GMRES method to solve multistage nonlinear MPC problems, specify the
Optimization.Solver property of your multistage nonlinear MPC object
"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 is
Display— Level of display. This is a string or character vector specified as either
"iter"to display solver information at every iteration, and use
"none"to suppress any solver display. The default value is
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 is
MaxIterations— Maximum number of iterations allowed. This is a positive integer representing the maximum number of inner-loop iterations allowed for the solver. The default value is
Restart— Number of outer-loop iterations. This is a nonnegative integer representing the maximum number of outer-loop iterations for the solver. The default value is
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.
StabilizationParameteris too large, it may cause overcorrection and oscillations, leading to slow convergence or even divergence. On the other hand, if the
StabilizationParameteris too small, it may result in slow convergence as the step size is overly conservative. To enhance algorithm convergence, it is advisable to select a
StabilizationParametervalue of 1/Ts, where Ts represents the model sampling time. This parameter should not exceed 2/Ts.
The default value is
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 is
For an example on how to use the C/GMRES solver, see Control Robot Manipulator Using C/GMRES Solver.
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 MAT-file 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
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:
NZ is the number of decision variables.
Mcineq is the number of linear inequality constraints.
Mceq is the number of linear equality constraints.
Ncineq is the number of nonlinear inequality constraints.
Nceq is the number of nonlinear equality constraints.
Nonlinear cost function to minimize, specified as a handle to a function with the signature:
[F,G] = FUN(z)
|Input||Initial guesses for decision variable values, specified as a vector of length NZ|
|Input||Linear inequality constraint array, specified as an
array. Together, |
|Input||Linear inequality constraint vector, specified as a column vector of length
Mcineq. Together, |
|Input||Linear equality constraint array, specified as an
array. Together, |
|Input||Linear equality constraint vector, specified as a column vector of length
Mceq. Together, |
|Input||Lower bounds for decision variables, specified as a column vector of length NZ, where .|
|Input||Upper bounds for decision variables, specified as a column vector of length NZ, where .|
Nonlinear constraint function, specified as a handle to a function with the signature:
[cineq,c,Gineq,Geq] = NLCON(z)
|Output||Optimal decision variable values, returned as a vector of length NZ.|
|Output||Optimal cost, returned as a scalar.|
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