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Ch.Sunil Kumar

M.E (Aerospace)

IISc Bangalore

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LMI (Linear Matrix Inequalities)

The PI and PID control problem in an aircraft application can be converted into Static Output

Feedback problem and then solved using LMI.

The advantage of using the LMI approach is that

Various problems like Synthesis of, H2, mixed sensitivity

H2/Hinfinitycontrollers, Quadratic performance computation, Pole Placement Constraintsand -analysis have equivalent LMI formulations.

Various iterative and non-iterative algorithms have been in use to solve these inequalities.

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Modified ILMI Algorithm In this algorithm, plant models for various trim conditions are considered.

The controller is found by solving the inequalities for a particular plant model.

The controller is verified with the other plant models for its stability.

The H2 norm condition is included in the inequalities and the Hinfinity norm, Gain

and Phase Margin requirements are verified.

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Probabilistic Robust Control The classical Robust Control techniques suffer from either over conservativeness or from

computational intractability.

The Robustness in probabilistic sense ensures that the certain property of a control system

(stability or the performance ) is said to be robust if it holds for most of the instances of

uncertainty.

The exact computation of multi dimensional probability density function is computationally

intractable problem.

The performance probability is estimated by randomly sampling the uncertainties.

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Statistical Learning theory In this theory, the goal of the algorithm is to find a controller such that the average

performance (with respect to uncertainty ) of the controller is minimized.

The objective is to find 0 such that the optimal average performance where is therange of variation of controller parameter variation.

The minimization of the average performance is different from the usual Robust design

method where worst case performance is minimized. Thus the controller does not produce an over conservative design and also it is

computationally efficient.

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Algorithm

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The cost function is defined as

where we sample P(x) and C(y) from the

Plant and controller space respectively

The controller is of the form

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Combining the Randomized algorithm with ILMI

approach

The plant space is plant models at various trim conditions with uncertainty present in themodels.

The Controller space is the space occupied by the controller which satisfies the Gain

Margin , Phase Margin Requirements. The Controller space is not known beforehand. The Plant space and the Controller space looks like

Plantspace

Controllerspace

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The cost function is defined as

The ILMI algorithmgives the controllerwhich is checked for

the Gain and Phasemargin requirements.

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Plantspace

Controllerspace

If we consider five plants at five different trim conditions we get 5different controllers.

From the 5 controllers so obtained we can get the Controller spacedepending on the variation of coefficients of the controller.

e.g. If the controller is of the form (c1*s)/(s+c2) then we can get the

range of variation of c1,c2 and this forms the controller space.

From the Controller and plant space we can apply Randomizedalgorithm to sample the plant and controller space and find thecontroller.

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