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Tutorial

This tutorial will be updated soon, and will be replaced step-by-step with tutorial-style blog posts. With the release of OpenOCLv3 some things have changed. To get the latest class and function signatures please see the API docs. As there is also a new style of implementing systems and optimal control problems, have a look the new example.

In this guide you will learn how to implement a system model and an optimal control problem with OpenOCL. You can open the linked examples in separate tabs in your browser (holding down ctrl). Also check out the API docs for a more detailed vision into the function’s signatures.

Defining a system model

Have a look at the CartPoleSystem.m in the Examples folder. The system is implemented by inheriting from the OclSystem class. You need to implement the two methods setupVariables and setupEquation.

setupVariables: With the setupVariables method you can define the system variables. You can create state, control and algebraic variables using the methods: addState, addAlgVar, addControl. They have the following signature with no return values:

self.addState(id,size)  
self.addAlgVar(id,size)
self.addControl(id,size) 

Every variable needs to have a unique string valued id. The size is given as a 2 dimensional matrix e.g. self.addState('p',[3,1]) to create a state variable p of size [3,1]. The size is optional for a scalar variable or can be a scalar value to create a variable with the given number of elements.

setupEquation: With the setupEquation method you can define the system equations. Ordinary differential equations (ODE) and differential algebraic equations (DAE) have to be stated in explicit or semi-explicit form. The signature of the method is:

function setupEquation(self,states,algVars,controls,parameters)

The setupEquation method takes 4 input arguments of type Var: states, algVars, controls, parameters. The Var class provides useful methods to access the content of variables (see description of Var below). For example, you can get the value of the 3rd position state variable p by: p = states.get('p'); (shorthand notation: p = states.p;)

Inside setupEquation, you can use the self.setODE method to specify the derivative for each state variable The setODE method has the following signature with no return value

self.setODE(id,equation)

For example to define the ODE for the positions p variable by: self.setODE('p',v); where the velocity v shall be of the same dimension as p.

Use the self.setAlgEquation method to define the algebraic equations. It has the following signature with no return value

self.setAlgEquation(equation) 

The algebraic equation needs to be given as a column vector. You can use the setAlgEquation method multiple times. In order to be able to simulate the system, the total number of rows of the algebraic equations needs to be equal to the total number/dimension of algebraic variables (number of degrees of freedom).

Defining an optimal control problem (OCP)

Have a look at the CartPoleOCP.m in the Examples folder. The OCP is implemented by inheriting from the OclOCP class. While the setupEquation method for the OclSystem is mandatory, the methods in OclOCP to define the cost functions and boundary conditions are optional. Default values for the costs are zero and an empty constraint list for the constraints.

Cost functions: pathCosts, arrivalCosts In the pathCosts method you can implement the intermediate cost (Lagrange costs) function, and in the arrivalCosts (Mayer costs) method you implement the costs on the final state.

function pathCosts(self,states,algVars,controls,parameters)  
function arrivalCosts(self,states,parameters)  

The variables states, algVars, controls, and parameters are of type Var. Use the functions addPathCost and addArrivalCost to add a cost term in the respective methods.

self.addPathCost(equation)  
self.addArrivalCost(equation)

Path constraints: Use the pathConstraints method for implementing path constraints The signature is (no return value):

function pathConstraints(states, algVars, controls, parameters)  

Inside pathConstraints you can use the function addPathConstraint to add a constraint, the signature is:

self.addPathConstraint(lhs, operator, rhs)   

where lhs and rhs are the left hand side and the right hand side of the equation and the operator can be one of ‘<=’, ‘>=’, ‘==’

Boundary conditions: You can add boundary conditions on the initial and the final state of the trajectory. The signature is:

function boundaryConditions(self, states0, statesF, parameters)  

You can use the functions addBoundaryCondition to add a constraint, the signature is:

self.addBoundaryCondition(lhs, operator, rhs)   

where lhs and rhs are the left hand side and the right hand side of the equation and the operator can be one of ‘<=’, ‘>=’, ‘==’

Calling the solver

The essential steps in order to solve your dynamical optimization problem are:

Have a look at the Example script to get an idea how it works: mainCartPole.m

The Variable class: Accessing variables, initial guess, and solution

The Variable type(or CasadiVariable in the CasADi backend) is the basic structure to retrieve, store, modify structured optimization variables.

Accessing values

For each variable you can access specific variables by their name, e.g. you can get the position variable from the state by:

states.get('p')

or

states.p

If there are multiple variable with the same name, you can slice the variables using a second parameter:

initialGuess.get('states').get('p')  % or initialGuess.states.p will return the position trajectory but   
initialGuess.get('states',1).get('p') % will return only the first position of the trajectory or use   
initialGuess.get('states').get('p',1) % to receive the trajectory of the first position coordinate.

The get operator always returns another Algorithmic object. To perform mathematical operations on objects of type Var, you need to access the value of the variable with the value attribute. This can also be useful for plotting when standard Matlab variables types are required:

states.get('p').value

or

states.p.value

Setting values

You can set values to variables using the set methods. The signature is:

var.set(value)

For example to the set the position trajectory in the initial guess, you can use:

initialGuess.get('states').get('p').set(positionTrajectory)   

The dimension of the value has to match the size of the variable with the following exceptions. If the value is scalar, then the same value will be set to all entries of the variable. If the number of elements of the value coincides with the number of entries of the variable, then the values will be assigned in the order of the variable (when you assign with e.g. with a column vector value).

initialGuess.get('states').get('p').set(0)  % will set all positions variables to [0;0;0].

Additional Information