Controller Design for Multivariable Systems Via Engineering Performance Indices. Part I

This first part of the article discusses the design of an output feedback controller for a multivariable control system. The controller is designed to provide specified or achievable performance in terms of: control errors, stability margins, and response time. The plant is subject to bounded external disturbances. The design problem is solved using a modified H∞-optimization procedure. The second part of the article provides an interpretation of the stability margin radii for a multivariable system, formulated in terms of Nyquist plots with breaking points at individual plant inputs. A direct relationship is established between the absolute stability of a closed-loop multivariable system with sector nonlinearities at the plant’s input and its stability margin radii. The proposed approach is illustrated by an example of controller design for a load-interconnected electric drive, demonstrating its relevance to engineering practice. This paper addresses the synthesis of output controllers for linear multivariable systems, using performance indicators that are widely adopted in engineering practice. These will be referred to as engineering quality indicators. These indicators are used in practice to assess the effectiveness of a designed closed loop control system. It is essential to note that the engineering quality indicators admit experimental verification. These indicators characterize the system’s accuracy (assessed by control errors under bounded disturbances), its response time (determined by the settling time), and its stability margins (evaluated via Nyquist plots of the open-loop system for each measured and control variable, i.e., at the plant’s physical output and input). In the classical automatic control theory for scalar systems, stability margins are typically quantified by gain and phase margins. However, in some cases, these indices can be misleading; the Nyquist plot of the open-loop system may pass very close to the critical point (–1, j0) without a corresponding significant reduction in their values. Therefore, this article employs the stability margin radius defined as the minimum distance from the critical point (–1, j0) to the Nyquist plot to assess robustness. This radius allows for the direct specification of guaranteed lower bounds on the classical gain and phase margins. This first part of the work considers multivariable systems subjected to unmeasured, bounded external disturbances. The disturbances are assumed to be continuous and piecewise differentiable, a class that covers most scenarios encountered in engineering practice. The objective is to synthesize an output controller that guarantees specified or achievable engineering quality indices: control errors for each output, settling time, and stability margins evaluated at the plant’s physical input. Moreover, a dedicated stability margin radius is guaranteed for each individual control input. The synthesis problem is solved via a modified standard H∞ optimization procedure. The order of the synthesized controller does not exceed that of the plant. The operating accuracy requirements are satisfied by selecting a diagonal weight matrix for the controlled variables in the optimization criterion. The matrix elements are determined via rigorous formulas that utilize the known amplitudes of disturbances and the specified control errors. The second part of this work analyzes the frequency-domain properties of the synthesized system using Nyquist plots of the loop transfer function, broken at each individual plant input. The absolute stability of the closed-loop system with sector nonlinearities at the plant input is proven. The size of the sector is directly determined by the achieved stability margin radius for that input. An example illustrating the method’s effectiveness is provided, based on the synthesis of a multivariable output controller for a practical electromechanical system.

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