configure optimization solver for nonlinear mpc -凯发k8网页登录

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. if you do not have optimization toolbox software, you can specify your own custom nonlinear solver.

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 p-1}. 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 is 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:

return the opt output argument when calling . this object 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 (opt.slack0).
pass this object in as the options input argument to nlmpcmove for the next control interval.

these command-line simulation steps are best practices, even if you do not specify any other run-time 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 (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.

specify custom solver

if you do not have optimization toolbox software, 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 .

you can use the nonlinear programming solver developed by to simulate and generate code for nonlinear mpc controllers. for more information, see .

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: `z` — decision variables, specified as a vector of length n_z. `f` — cost, returned as a scalar. `g` — cost function gradients with respect to the decision variables, returned as a column vector of length n_z, where $g (i) = \partial f / \partial z (i)$ .
`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-by-n_z array. together, `a` and `b` define constraints of the form $a \cdot z \leq b$ .
`b`	input	linear inequality constraint vector, specified as a column vector of length m_cineq. together, `a` and `b` define constraints of the form $a \cdot z \leq b$ .
`aeq`	input	linear equality constraint array, specified as an m_ceq-by-n_z array. together, `aeq` and `beq` define constraints of the form $aeq \cdot z = beq$ .
`beq`	input	linear equality constraint vector, specified as a column vector of length m_ceq. together, `aeq` and `beq` define constraints of the form $aeq \cdot z = beq$ .
`lb`	input	lower bounds for decision variables, specified as a column vector of length n_z, where $lb (i) \leq z (i)$ .
`ub`	input	upper bounds for decision variables, specified as a column vector of length n_z, where $z (i) \leq ub (i)$ .
`nlcon`	input	nonlinear constraint function, specified as a handle to a function with the signature: [cineq,c,gineq,geq] = nlcon(z) and arguments: `z` — decision variables, specified as a vector of length n_z. `cineq` — nonlinear inequality constraint values, returned as a column vector of length n_cineq. `ceq` — nonlinear equality constraint values, returned as a column vector of length n_ceq. `gineq` — nonlinear inequality constraint gradients with respect to the decision variables, returned as an n_z-by-n_cineq array, where $gineq (i, j) = \partial cineq (j) / \partial z (i)$ . `geq` — nonlinear equality constraint gradients with respect to the decision variables, returned as an n_z-by-n_ceq array, where $geq (i, j) = \partial ceq (j) / \partial z (i)$ .
`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: `1` — optimization successful (first order optimization conditions satisfied) `0` — suboptimal solution (maximum number of iterations reached) negative value — optimization failed

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 .

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