Algebraic Variable

[Adam Makin]

Thanksagain for the quick response. I kept getting the same error message despite following your suggestions. After reading the manual, I was wondering if should I use IRK instead of ERK (since I use algebraic states).

I just realized that currently only the linear least squares cost module can handle the algebraic variables.

The Nonlinear least squares cost module in C ( _nlp/ocp_nlp_cost_nls.c) has to be adapted to handle also algebraic variables.

This is a bit more complicated then adding it to the code generation.

In case that your measurement function h(x,z) is linear, you can actually use the LINEAR_LS cost.

You could also implement h as a slacked constraint and penalize the corresponding slacks with the same weights, which should be equivalent but admittedly is not as clean.

Thanks a lot for all the help! After your hints on how to transcribe the current example to the LINEAR_LS formulation, I could get the code to compile and iterate. Sadly, I always get zero as a result.

Q0_x and Q0_z are also identity matrix. simY has the same data that I also use with my CasADi + IPOPT MHE solver. Acados returns status 0. Yesterday I changed the print_level option from 0 to 1 and I found out that the SQP only iterates one time (returns at iteration 0).

If the evolution of these variables is not defined by such an equation, the solver can freely choose it and the u control discretization should be appropriate.

The variable is constant over the last shooting interval and can be penalized at the N-1 node.

Last night I reformulated the problem and changed the algebraic variables to control inputs by doing u = vertcat(w, z) (I keep the variable name z here although it might be misleading). After making these changes, the MHE solver works and the result are consistent with the results obtained using CasADi + IPOPT.

I mean, that what you get with

acados_solver_mhe.get(N-1, "u")

is the value which is valid for the last shooting interval, i.e. the time horizon between shooting node N-1 and N.

So if you want to just get the value, this should be good enough.

Or why do you need it at the Nth node?

If you want to penalize the last value of u (your concatenation) differently, you can just set the weighting matrix at shooting node N-1 accordingly.

Is the auto generated main*.c file good to use as a starting point for the MHE example or should I take something else in account? It seems that I am doing a different thing in Python to what is done in the autogenerated C file.

I think it is a good starting point, since it shows how the solver can be used from C.

Note that the main*.c file just creates the solver (in your case MHE) and initializes it with some values from the json file.

The interaction with the solver, i.e. all acados_solver_mhe.set(*), acados_solver_mhe.get(*) kind of calls have to be translated from the Python interface into acados C interface calls.

I guess MHE is only meaningful when the solver interacts with an integrator/OCP solver, which is not done in the generated main file.

The term variable means "to change". This is the reason why we usealgebraic variables. If a number can change based on the situation,then we would use a variable in it's place. This is better representedwhen we talk about formulas. Take a look....

In mathematics, a variable (from Latin variabilis, "changeable") is a symbol that represents a mathematical object. A variable may represent a number, a vector, a matrix, a function, the argument of a function, a set, or an element of a set.[1]

Algebraic computations with variables as if they were explicit numbers solve a range of problems in a single computation. For example, the quadratic formula solves any quadratic equation by substituting the numeric values of the coefficients of that equation for the variables that represent them in the quadratic formula. In mathematical logic, a variable is either a symbol representing an unspecified term of the theory (a meta-variable), or a basic object of the theory that is manipulated without referring to its possible intuitive interpretation.

In ancient works such as Euclid's Elements, single letters refer to geometric points and shapes. In the 7th century, Brahmagupta used different colours to represent the unknowns in algebraic equations in the Brāhmasphuṭasiddhānta. One section of this book is called "Equations of Several Colours".[2]

In the second half of the 19th century, it appeared that the foundation of infinitesimal calculus was not formalized enough to deal with apparent paradoxes such as a nowhere differentiable continuous function. To solve this problem, Karl Weierstrass introduced a new formalism consisting of replacing the intuitive notion of limit by a formal definition. The older notion of limit was "when the variable x varies and tends toward a, then f(x) tends toward L", without any accurate definition of "tends". Weierstrass replaced this sentence by the formula

This static formulation led to the modern notion of variable, which is simply a symbol representing a mathematical object that either is unknown, or may be replaced by any element of a given set (e.g., the set of real numbers).

Specific branches and applications of mathematics have specific naming conventions for variables. Variables with similar roles or meanings are often assigned consecutive letters or the same letter with different subscripts. For example, the three axes in 3D coordinate space are conventionally called x, y, and z. In physics, the names of variables are largely determined by the physical quantity they describe, but various naming conventions exist. A convention often followed in probability and statistics is to use X, Y, Z for the names of random variables, keeping x, y, z for variables representing corresponding better-defined values.

In the same context, variables that are independent of x define constant functions and are therefore called constant. For example, a constant of integration is an arbitrary constant function that is added to a particular antiderivative to obtain the other antiderivatives. Because of the strong relationship between polynomials and polynomial functions, the term "constant" is often used to denote the coefficients of a polynomial, which are constant functions of the indeterminates.

In calculus and its application to physics and other sciences, it is rather common to consider a variable, say y, whose possible values depend on the value of another variable, say x. In mathematical terms, the dependent variable y represents the value of a function of x. To simplify formulas, it is often useful to use the same symbol for the dependent variable y and the function mapping x onto y. For example, the state of a physical system depends on measurable quantities such as the pressure, the temperature, the spatial position, ..., and all these quantities vary when the system evolves, that is, they are function of the time. In the formulas describing the system, these quantities are represented by variables which are dependent on the time, and thus considered implicitly as functions of the time.

Therefore, in a formula, a dependent variable is a variable that is implicitly a function of another (or several other) variables. An independent variable is a variable that is not dependent.[9]

The table has two rows. The top row shows the random variable for each of the 6 possible credit points: X = 1, 2, 3, 4, 5, and 6. The second row beneath shows the probability for each credit point P(x) is one/sixth for each variable above.

Using standard formulas, students encounter expressions like x = 3.5 and , which use X in this way. Without much experience with these types of variables, it often takes some time for students to figure out what the various symbols actually represent.

Back to our hypothetical: As the semester wears on, some students are not completing all the work. So, as an extra incentive late in the term, the teacher decides to offer a double bonus, between 2 and 12 points. There are now two different ways that the teacher can proceed: Either have the student roll one die, then double the value, or roll two dice, and add the sum as bonus points.

Are these procedures any different from each other? They have the same maximum and minimum values (2 and 12) and the same mean (7). However, they neither have the same probability distribution nor the same variability. In the first case, there are only six possible outcomes when doubling the value of one die: the even numbers from 2 to 12. In the second case, all the values from 2 to 12 might result from the random process.

The table has two rows. The top row shows the doubling of the random variable for each of the 6 possible credit points as: D = 2, 4, 6, 8, 10, and 12. The second row beneath shows the probability for each credit point P(D) remains one/sixth for each variable above.

The outcomes are uniformly distributed because all outcomes have the same probability of occurring. Again, employing the standard formulas D = 7 and , how do these compare to the values for rolling a single die, X?

As you introduce your students to random variables, be ready to clear up confusion regarding ideas they are accustomed to, algebraic variables, and the new ideas. Provide your students with settings in which they can work with random variables, write expressions using random variables, and gain the intuition that they need if they are to use these ideas effectively in their work in statistics.

In Mathematics, an Algebra is a branch that deals with symbols, variables, numbers and the rules for manipulating it. It states the mathematical relationship are used to find the unknown value by creating expressions and equations. Now, consider the following algebraic expression,

All algebraic expressions and terms consist of at least one variable. It is the variable which distinguishes an algebraic expression from an arithmetic one. The presence of a variable in a mathematical expression enables infinite possibilities to determine the value of the expression.

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