Minimal-norm globally non-overshooting/undershooting control of linear multivariable systems(MS)

Abstract

One of the key objectives to consider while designing closed-loop control systems is that the plant output should follow the pre-specified reference signal or the setpoint without any overshoots or undershoots. Accordingly, the literature encompassing the broad area of control theory has extensively addressed the transient response improvement and tracking control problems of linear time-invariant (LTI) systems. Several design techniques exist in the literature for the transient response shaping of single-input/output (SISO) LTI systems utilising state-feedback controllers. While the state-feedback controller design methods for SISO systems can be suitably utilised for multi-input/output(MIMO) systems to non-uniquely place the eigenvalues at the desired locations, eigenvalue assignment alone may not achieve the desired transient performance for MIMO systems due to the coupling between the input and output variables. For several “positioning” control problems such as the control of elevators, hard disk read/write head control, coordination of multiagent systems, etc., achieving the non - overshooting/ undershooting (NOUS) tracking control objectives in the output response of the closed-loop system are crucial. These objectives simultaneously target both the steady-state and transient performance specifications and are achieved by assigning the complete eigenstructure (eigenvalues and eigenvectors) of the resulting closed-loop system. The “globally” NOUS (GNOUS) control problem refers to obtaining the response from a state-feedback control system to a constant step reference signal such that the resulting system yields a monotonic response, independently of the initial conditions and step reference magnitudes. In addition, one of the desired prerequisites of an industrial plant is to have a minimum operation and maintenance cost, which can be achieved by installing low-cost actuators. Mathematically, this translates to guaranteeing the desired control performance with the minimal control efforts, or equivalently, synthesizing the state-feedback controller of the minimal Frobenius norm. The above two control objectives, namely, GNOUS tracking and minimal norm of the state-feedback gain matrix, are widely addressed in the literature. Since these control objectives are conflicting because they are inversely proportional to each other, particularly for the case of scalar input LTI systems, they have been addressed separately even for multiple input LTI systems. Through this work, the open problem of simultaneously achieving these set of objectives for multivariable LTI systems is formulated and extensively addressed. Firstly, new results for the GNOUS tracking control problem for MIMO LTI systems are presented, incorporating the practically relevant control objective of a minimal-norm feedback gain matrix. Further, an algorithm is presented to synthesise a minimalnorm state-feedback controller such that the GNOUS tracking control objectives are achieved in all the output responses of the closed-loop system with the desired rate of convergence. The key idea behind this algorithm is to utilise the additional degree of freedom of multivariable systems for placing the eigenvalues of the closed-loop system within a pre-specified region in the complex plane. An upper bound on the worst-case convergence rate in all output responses for a pre-specified disk region is also provided. Subsequently, the proposed algorithm is extended to the class of decoupled largescale MIMO LTI systems. These systems feature a high-dimensional plant model and are composed of a number of independent subsystems. Multi-area power systems, gas distribution networks, and ecological systems are all examples. Finally, this work demonstrates some numerical and experimental results to validate the effectiveness of the proposed algorithms.