MIMOtool Help Desk

H2 & H-INFINITY

(Output Feedback)

For the H2 and H¥ problem formulation, it's necessary to consider the following linear fractional transformation (LFT) on the closed loop system [7]

which can be represented by the diagram

where the signal w(s) contains all the external input (noises, disturbances, references), the output z(s) is the error signal, y(s) are the measured variables and u(s) is the control input; P(s) is the generalized plant (current model plus all the necessary weighting functions) and K(s) is the controller (Kof plus the blocks of integrators which, if necessary, have been added to the current model).

The H2 and H¥ problem [8] is to find, among all controllers that yields an internally stable closed loop system, a controller K(s) that minimizes respectively the root mean square (2-norm) of the output z(s) of the plant P(s) with white noise in input (H2 problem) and the maximum singular value over frequency (infinity-norm) of Pzw(s) (H¥ problem).

The design specifications can be resumed in the following remarks:

· At low frequencies, where a satisfying knowledge of the plant is assumed, the most important requirements are the tracking of the reference signal and a good disturbance rejection, which can be both meet by keeping the Output Sensitivity low:

s_max( So(jw ) ) << 1

· At high frequencies, the most important requirements are sensor noise rejection and robust stability in face of uncertainties due to unmodelled dynamics, non linearities and system truncation, which usually become marked as the working frequency increases; moreover the control energy has to be reduced as much as possible at these frequencies, since no reference signal must be followed. These requirements can be meet keeping the Output Control Sensitivity or the Output Complementary Sensitivity as low as possible (note that if M(s) is low, the control signal u(s) and hence the matrix T(s) are low too):

s_max( Mo(jw ) ) << 1 or s_max( To(jw ) ) << 1

If the functions WM(s)=1/Mmax(s), WS(s)=1/Smax(s) and WT(s)=1/Tmax(s) express the desired shape for Mo, So and To, the conditions above are satisfied if:

s_max( So(s) WS(s) ) < 1

s_max( Mo(s) WM(s) ) < 1 or s_max( To(jw ) WT(s) ) < 1

Then, the first step in H2 and H¥ synthesis is the choice of the matrices which have to be minimized in order to construct the relative standard plant P(s) in such a way that the transfer matrix Pzw(s) includes the functions used to express the conditions:

· If the user wants to minimize the matrix [ To(s) ; So(s) ],

Pzw(s) = [ To(s)WT(s) ; So(s)WS(s) ]

· If the user want to minimize the matrix [ Mo(s) ; So(s) ],

Pzw(s) = [ Mo(s)WM(s) ; So(s)WS(s) ]

The shapes of the weighting functions WS and WT (or WM), and hence the upper limit of the corresponding transfer matrices So and To (or Mo), can be modified in the following window by means of the two parameters X1 and X2:

The button PLOT reads X1 and X2 and, if they are valid, displays on the graphic the shapes of the upper limits, while the button NEXT stores the two parameters and opens the window dedicated to the computation of the controller:

The two radio buttons of the first frame allow to choose the way in which the plant has to be weighted: only the use of a derivative action permits an exact weighting [9] of the Control or Complementary Sensitivity, otherwise an approximate one could be applied.

The other frame is dedicated to the selection of the function from which the controller K(s) has to be obtained and the button COMPUTE H-... starts this computation; the end of the calculation is pointed out with a text note appearing on the window and, if the controller has been found, the two buttons EVALUATION and SIMULATION are activated.

Note: depending on the selected function, especially if it belongs to the LMI Toolbox, the controller computation could take a rather long time; in any case it's possible to visualize all the calculation steps on the Matlab window.