Control system design and analysis using closed-loop Nyquist and Bode arrays

In this paper a method is described for designing linear multivariable control schemes which have a closed-loop frequency response as close as possible, in a least squares sense, to a desired response. After using characteristic gain loci to ensure system stability, the closed-loop Bode array gives easily understood information about the controlled system in terms of bandwidth, speed of response, resonance and interaction. The closed-loop Nyquist array indicates the robustness of the control scheme for sensor failures ; it also indicates the extent to which state and input noise will be suppressed, since the feedback just multiplies the open-loop disturbances by a unit matrix minus the closed-loop frequency response. Bands of Gershgorin and Ostrowski circles are used to indicate the behaviour for changes in the characteristics of more than one sensor at a time. A similar frequency-response array, obtained by breaking the feedback loops next to the actuators instead of next to the sensors, can be used to predict the behaviour of the controlled system for actuator failures. Parameter sensitivity can be investigated by determining the rate of change of the closed-loop frequency response with changes in the parameter concerned. Three examples are used to illustrate these closed-loop array methods. Linear multivariable control schemes are designed for a 5-input, 5-output, 33-stato jet engine ; and for a 3-input, 2-output, 29-state chemical reactor. A parameter-dependent compensator is designed for a 2-input, 2-output, 8-state missile in which the control scheme was required to work for a wide range of values for one of the system parameters.