Model Based Predictive Control

The notion of Model based Predictive Control (MPC) is one that bases its control strategy on a known model of the output of the controlled system. By pre-emptively calculating the expected behaviour of a system, a controller is able to take appropriate control action based on that prediction (Åström & Hägglund, 1995). According to Merabet & Gu (2010), MPC is considered a very effective control method for non-linear processes and disturbances.

As stated by Åström & Hägglund (1995), there are three main steps to implementing an MPC controller, once a model of the system has been specified. The first step simply acquires information about the system, including the measure of process inputs, outputs and disturbances. From there, the current output and predicted future state of the system is calculated through use of the output model. Step two then calculates the control changes necessary to minimise error between the desired and actual trajectories, subject to prescribed constraints. The final step simply executes this control action and restarts the process.

There are several major advantages associated with MPC over traditional control methodologies according to Åström & Hägglund (1995):

x Multivariable

x Permits hard constraints x Is predictive

x Deals with trajectories rather than single points

In a paper published by Merabet & Gu (2010) in 2010, the fundamentals and implementation of a comprehensive MPC controller is evaluated for advanced non-linear robotic manipulator control.

The use of MPC to control a rigid robotic arm is somewhat problematic when compared to applying the same methodology on other systems. The increased control difficulty arises from un-modelled dynamics such as variable payloads, friction torques, torque disturbances, parameter variations and measurement noise (Merabet & Gu, 2010). Merabet & Gu (2010) suggest that this may be further emphasised by inaccuracies present in the robot model.

In order to implement MPC, first a mathematical model of the arm must be constructed with respect to angular joint positions. Merabet & Gu (2010) uses the Euler-Lagrange equation for a dynamic rigid robot manipulator of -links as a base for further development:

( ) ̈ + ( , ̇ ) ̇ + ( ) =

Where:

( ) ∈ ℜ angular joint position vector

( ) ∈ ℜ driving torque vector (control input)

( ) ∈ ℜ × _{, ( ) = ( ) > 0 link inertia matrix}

( , ̇ ) ̇ ℜ Coriolis & centripetal torques vector

( ) ∈ ℜ gravitational torques vector

When implementing an arm, there are practical uncertainties that must be considered and compensated for to achieve robust and accurate control. These may include factors such as modelling errors, unknown loads and computational errors (Merabet & Gu, 2010).

Uncertainty added to matrices:

( ) = ( ) + ∆

( , ̇ ) = ( , ̇ ) + ∆

( ) = ( ) + ∆

∆(. ) system error

Figure 2 - 91. Uncertainty definitions. (Merabet & Gu, 2010).

Adding uncertainty and other unmodelled quantities to robot equation:

( ( ) + ∆ ) ̈ + ( ( , ̇ ) + ∆ ) ̇ + ( ) + ∆ + = +

( ) ∈ ℜ friction vector

( ) ∈ ℜ external disturbance vector

Figure 2 - 92. Manipulator arm equation with added uncertainty. (Merabet & Gu, 2010).

Simplification:

( ) ̈ + ( , ̇ ) ̇ + ( ) = + ƞ( ̈ , ̇ , , )

ƞ ( ) − ∆ ̈ + ∆ + ∆ +̇ −

Figure 2 - 93. Uncertainty definition for simplification. (Merabet & Gu, 2010).

Robust control law:

( ) = − ( ) ( − ) + ( ̇ − ̇ ) − ( ) ( , ) + ( ) − ̈ − ƞ ( )

10

3 ∗ ×

5

2 ∗ ×

Figure 2 - 94. Robust control law defined by Merabet & Gu (2010).

This produces the final dynamic nonlinear mathematical model of a robotic arm as prescribed by Merabet & Gu (2010) in terms of the state vector :

̇ = ( ) + ( ) + ( )ƞ

( ) _{− ( ) ( , ) + ( )}

( ) 0 ×

( )

Figure 2 - 95. Final dynamic nonlinear model of robotic manipulator. (Merabet & Gu, 2010).

= ℎ( ) =

[ × 0 × ]

Figure 2 - 96. Output vector definition. (Merabet & Gu, 2010).

According to Merabet & Gu (2010), the associated control law must be derived from an undisturbed system where uncertainties are excluded. When the appropriate model has been achieved, robust control law follows and is mathematically implemented by compensating for any error through estimation.

Once a robust mathematical model of the manipulator arm to be controlled has been constructed in terms of measureable and controllable parameters, MPC may be employed. The supposed control system as defined by Merabet & Gu (2010), consists of an optimisation algorithm to solve for the future control trajectory through the use of the dynamic process model (fig. 2 - 95). It aims to do this through the minimisation of a cost function.

General form of cost function:

ℑ = ( + ), ,

( + ) predicted error

( + ) -step ahead prediction

> 0 prediction time horizon

Figure 2 - 97. General form cost function definition. (Merabet & Gu, 2010).

Merabet & Gu (2010) then uses a mathematical tool based on Taylor series expansion carried out by Lie derivatives to develop the prediction model, by use of the process model. Yielding a final prediction model of:

( + ) = ( ) ( ) ( ) ⎣ ⎢ ⎢ ⎡ _∗ × × 2 ∗ × ⎦ ⎥ ⎥ ⎤ ( ) = ( ) ̇( ) ̈( ) − ( ) ( , ) + ( ) + 0 × 0 × ( ) ( )

Figure 2 - 98. Final predictive model. (Merabet & Gu, 2010).

Similarly, a reference trajectory analysis may be constructed as follows:

( + ) = ( ) ( )

( ) [ ̇ ̈ ]

Allowing for the predicted error calculation required by the cost function:

( + ) = ( + ) − ( + ) = ( )( ( ) − ( ))

Figure 2 - 100. Predicted error calculation. (Merabet & Gu, 2010).

It is now possible to carry out optimal control law, through the minimisation of the cost function in relation to the control input (Merabet & Gu, 2010). At this point, the definition of the cost function is crucial. Merabet & Gu (2010) has provided two approaches:

ℑ =1

2 ( + ) ( + ) +

1

2 ( + ) ( + )

Figure 2 - 101. Expanded general form of cost function. (Merabet & Gu, 2010).

ℑ =1

2 ( + ) − ( + ) ( + ) − ( + )

Figure 2 - 102. Tracking error-based cost function. (Merabet & Gu, 2010).

To evaluate the transient response and robustness of the suggested control, Merabet & Gu (2010) mathematically simulate the previously described system on a rigid robot manipulator of two links. The controller is of form fig. 2 - 103.

Figure 2 - 103. Model based predictive control diagram. (Merabet & Gu, 2010).

Model simulation was implemented with a sample time of 100 seconds (Merabet & Gu, 2010), with the manipulator initially stationary. An external disturbance was applied periodically. Merabet & Gu (2010) suggests robust control of the simulated manipulator has been achieved, providing the resultant angular positions and tracking errors for both controlled joints as evidence (fig. 2 - 104). As a side note, it is mentioned that the manipulator presents periodic steady state errors, brought on by external disturbances. It is also stated that this may be mitigated if this information was made available to the control law. Further control improvement may be implemented if load parameters are known (Merabet & Gu, 2010); in this case, it is regarded as part of the second link, with the system accommodating the load through a change in link properties such as mass, length, centre of mass and moment of inertia.

Figure 2 - 104. Controller joint set-point response diagrams. (Merabet & Gu, 2010).

Merabet & Gu (2010) have concluded that this control system through investigation has proved able to provide robust control with respect to modelling errors, effective disturbance rejection and no parameter uncertainty induced steady state error. However, Merabet & Gu (2010) state that it is well known that a theoretical model of a systems behaviour may not always translate well into reality. The uncertainty compensator is one approach of dealing with this issue. Merabet & Gu (2010) suggest that a Fuzzy logic controller may be another viable alternative.