Mathematica assistance in proving theorems in nonlinear control

Ülle Kotta & Palle Kotta

Institute of Cybernetics at Tallinn University of Technology, Estonia

Control system

Control systems are systems that have at least one variable that  can be changed (within certain limits) in order to control other system’s variables.

Example

A simple example of control system is TV remote control. 

We can change input variable (by pushing the channel button on the remote) in order to control output variable (channel on TV).

Input-output vs state equations

Control system can be described in many ways, including:

input-output equations

Input-output equations describe the relation between system’s inputs and outputs, and possibly their derivatives.

Example

The channel shown on TV (output) is related to the button on the remote (input).

state equations

State is a function of inputs, outputs and their derivatatives.

From one n-th order differential equation (i/o equation) it is possible to obtain the system of 1st order differential equations (state equations).

State equations: example (pendulum)

input-output equations

(ml^2)(Overscript[φ, ..])=mgl sinφ=u

state equations

(Subscript[(OverDot[x]), 1])=(Subscript[x, 2])
(Subscript[(OverDot[x]), 2])=(1/(ml^2))(u+mgl sin (Subscript[x, 1]))
  φ=(Subscript[x, 1])

g: gravitational constant, l: length of pendulum, m: mass of pendulum, ϕ: angle of pendulum, u: torque

Model

In order to control processes more accurately and easily, we need a process (system) model.

Modelling

In modelling a system, we usually measure the inputs and outputs of the system and based on these measurements build up the input-output equations.

Input-output equations don’t usually come from first principles and may not correspond to the inner logic of the system.

Identification

Building an (input-output) equation based on measurements of system’s inputs and outputs, is called identification.

Why state equations?

Most control system related theory uses state equations.To apply control theory on systems to be controlled, we should obtain state equations from input-output equations.Obtaining state equations from input-output equations is called realization.

Realization

Obtaining state equations from input-output equations may sometimes be impossible.

It is good to know, if an input-output equation has a state equation.If we would knew that the model structure does not have state equation, we could choose (due to certain freedom) another similar input-output model structure to describe the system.

Linear control systems

From linear input-output equations, it is always possible to obtain state equations.

Linear control systems are simple but often not suitable for applications.

Nonlinear control system

Unlike linear systems, it is not always possible to find state equations for nonlinear systems.

Simple nonlinear control systems

In practice, most often two types of nonlinear control systems are used:

Bilinear

Quadratic

These two types of systems are easy enough to operate with and flexible enough to describe a system with desired accuracy.

Realizability

Realizability is a property of input-output equation which means it has state equations.

Modellers aim to describe system with realizable input-output equations in order to use tools from control systems theory.

Checking realizability

In the literature, there exists many methods for checking if the system is realizable or not.

Checking realizability and building state equation requires huge amount of complex symbolic computations.

Problem

Modellers should describe system with realizable input-output equations.

Modellers don’t know beforehand whether the input-output equations are realizable or not.

Checking the realizability takes much effort and time.

Our goal

Our goal was to obtain the subclasses of bilinear and quadratic input-output equations which are realizable, i.e.guaranteed to have state equations.

Using these subclasses, modeller could simply by direct inspection (without any calculations) check from the input-output equation parameters, if the system has state equations or not.

Tools used to obtain the results

We used theoretical results from the following book:

G.Conte, C.H.Moog, A.M.Perdon “Algebraic Methods for Nonlinear Control Systems.Theory and Applications”, 2nd Edition, Springer, 2007.

We used Mathematica together with nonlinear control system toolkit NLControl developed in our institute.

Mathematica assistance

Mathematica assisted finding the realizability conditions for bilinear and quadratic input-output equations.

Mathematica assisted finding the state equations for some simpler cases.

Mathematica assisted making hypothesis.

Mathematica assisted proving theorems.

Finding results: low order systems

Checking general realizability conditions requires solving a certain number (depending on the system order) of systems of nonlinear equations. For each system of equations there may exist many possible solutions. 

After finding all solutions for one system of equations, we omitted the solutions, which contained the other solutions. As a result, we got restrictions on system parameters ("branches"). For each set of restrictions, we repreated the process with the following system of nonlinear equations until the last system of nonlinear equations. 

Finally, after processing all “branches”, we omitted these sets of restrictions which were special cases of another sets of restrictions.

Finding results: subclasses

We found results for systems of order n.

We made hypothesis for order n + 1.

We checked the hypothesis for order n + 1.

After doing above steps for a few times, we made hypothesis for the general case.
  We proved the hypothesis for the general case theoretically.

Finding results: methods

We used two different methods for finding the necessary and sufficient conditions for bilinear input-output equations.

We used one method for computations on paper, and another one on computer algebra system Mathematica.

Both methods yielded the same results, as expected, but doing complex symbolic calculations on paper took more effort, and resulted in many intermediate mistakes.

Results (bilinear)

We found realizability conditions for 1st to 4th order bilinear input-output equations.

We found 2 general (of any order) realizable subclasses of bilinear input-output equations.

Bilinear input-output equation

Results for low order bilinear i/o equation

1st order – always realizable

2nd order – always realizable

3rd order – realizable for the cases

(Subscript[b, 1])=(Subscript[c, 11])=(Subscript[c, 21])=(Subscript[c, 31])=0

(Subscript[c, 11])=0, (Subscript[c, 21])=(Subscript[c, 12])

4th order

(Subscript[b, 1])=(Subscript[b, 2])=(Subscript[c, 11])=(Subscript[c, 12])=(Subscript[c, 21])=(Subscript[c, 22])=(Subscript[c, 31])=(Subscript[c, 32])=(Subscript[c, 41])=(Subscript[c, 42])=0

(Subscript[b, 1])=(Subscript[c, 11])=(Subscript[c, 12])=(Subscript[c, 21])=(Subscript[c, 31])=(Subscript[c, 41])=0, (Subscript[c, 13])=(Subscript[c, 22])

(Subscript[c, 11])=(Subscript[c, 12])=(Subscript[c, 13])=(Subscript[c, 21])=(Subscript[c, 22])=(Subscript[c, 31])=0, (Subscript[c, 23])=(Subscript[c, 32]),(Subscript[c, 14])=(Subscript[c, 41])

Results for general case bilinear i/o equation: 1st subclass

Results for general case bilinear i/o equation: 2nd subclass

Results (quadratic)

We found realizability conditions for 1st to 3rd order quadratic input-output equations.

We found 2 general (of any order) realizable subclasses of quadratic input-output equations.

Quadratic input-output equation

Conclusions

Although it is possible in principle to find the results for low order systems manually, probably it would not have been done.Computer algebra system makes  calculations much quicker and bypasses many human reliability issues.

While proving the conjectures on subclasses of arbitrary order was not very difficult theoretically, conjectures probably wouldn’t have been found without the aid of computer algebra system.

Thank you!