| In many physical systems, it is desirable to maintain operating conditions within small tolerances despite external disturbances. Presented is a controller design methodology for such systems which may be modeled as square, linear, uncertain plants subjected to an external disturbance. Specifically, each system output due to the disturbance is required to remain within a prespecified tolerance subjected to the constraint that each control effort is limited by its hardware. Like QFT, this methodology is based on the premise that feedback is demanded by plant uncertainty and external disturbances. Therefore, the amount of feedback used is contingent on the amount of uncertainty present in the system. The cost of using feedback to reduce system sensitivity and achieve disturbance rejection is reflected in the controller bandwidth which for all practical purposes must be minimized. The two key features in this design methodology are (i) the ability to incorporate controller constraints appearing as saturation bounds and bandwidth limitations and (ii) the utilization of loop interactions to satisfy the overall performance specifications of the closed loop system.; The output tolerances are satisfied through a modification of the QFT sequential technique presented in Yaniv (1986) which expresses the output transfer function matrix in a fixed point form. Likewise, the controller constraints are incorporated by expressing the controller closed loop transfer function matrix in a fixed point form. Although many fixed point expressions can be obtained for the control transfer function matrix, a useful expression has the characteristic of segregating the controllers such that each appears in its respective row thereby dividing the multivariable system into a family of {dollar}nsp2{dollar} SISO systems. The loop interactions naturally emerge as a consequence of employing the fixed point formulation where each performance tolerance represents its respective loop interaction. As the design proceeds, any previously designed loop information is incorporated into the remaining loops by utilizing the Gauss Elimination. The performance specifications are enforced by developing amplitude inequalities between the desired performance and pertinent closed loop transfer functions. Evaluating the output amplitude inequalities on the gain-phase plane gives a lower bound on the amplitude of the nominal loop transfer functions to be designed whereas evaluating the controller amplitude inequalities gives an upper bound on the amplitude of the nominal loop transfer functions. Hence, well-defined regions emerge confining the amplitudes of the nominal loop transfer functions as a function of phase. Once the design boundaries are obtained, standard loop shaping completes the design. |
