Subashish Datta

Partial Pole Placement with Controller Optimization

An arbitrary subset (n - m) of the (n) closed loop eigenvalues of an nth order continuous time single input linear time invariant system is to be placed using full state feedback, at pre-specified locations in the complex plane. The remaining m closed loop eigenvalues can be placed anywhere inside a pre-defined region in the complex plane. This region constraint on the unspecified poles is translated into a linear matrix inequality constraint on the feedback gains through a convex inner approximation of the polynomial stability region. The closed loop locations for these m eigenvalues are optimized to obtain a minimum norm feedback gain vector. This reduces the controller effort leading to less expensive actuators required to be installed in the control system. The proposed algorithm is illustrated on a linearized model of a 4-machine, 2-area power system example.

Partial pole placement with minimum norm controller

The problem of placing an arbitrary subset (m) of the (n) closed loop eigenvalues of a nth order continuous time single input linear time invariant(LTI) system, using full state feedback, is considered. The required locations of the remaining (n − m) closed loop eigenvalues are not precisely specified. However, they are required to be placed anywhere inside a pre-defined region in the complex plane. The resulting non-uniqueness is utilized to minimize the controller effort through optimization of the feedback gain vector norm. Using a variant of the boundary crossing theorem, the region constraint on the unspecified (n−m) poles is translated into a quadratic constraint on the characteristic polynomial coefficients. The resulting quadratically constrained quadratic program can be approximated by a quadratic program with linear constraints. The proposed theory is demonstrated for power oscillation damping controller design, where the eigenvalues corresponding to poorly damped electro-mechanical modes are critical for performance and hence are specified precisely by the designer, whereas the remaining eigenvalues are non-critical and need not be specified precisely. Acceptable closed loop pole placement is achieved for this example along with a 51% reduction in controller norm.

Feedback norm minimisation with regional pole placement

This article proposes a convex algorithm for minimising an upper bound of the state feedback gain matrix norm with regional pole placement for linear time-invariant multi-input systems. The inherent non-convexity in this optimisation is resolved by a combination of two separate approaches: (1) an inner convex approximation of the polynomial matrix stability region due to Henrion and (2) a novel convex parameterisation of column reduced matrix fraction system representations. Using a sequence of approximations enabled by the above two methods, it is shown that the constraints on closed-loop poles (both pre-specified exact locations and regional placement) define linear matrix inequalities. Finally, the effectiveness of the proposed algorithm is compared with similar pole placement algorithms through numerical examples.

Partial Pole Placement and Controller Norm Optimization over Polynomial Stability Region

Abstract An arbitrary subset ( n – m ) of the n closed loop eigenvalues of an n th order continuous time single input linear time invariant (LTI) system is to be placed using full state feedback, at pre-specified locations in the complex plane. The remaining m closed loop eigenvalues can be placed anywhere inside a pre-defined region in the complex plane. This region constraint on the unspecified poles is translated into an ellipsoidal constraint on the characteristic polynomial coefficients through a convex inner approximation for polynomial stability regions. The closed loop locations for these m eigenvalues are chosen through an explicit minimization of the feedback gain vector norm leading to an efficiently solvable semidefinite program. The required controller effort is thus minimized leading to less expensive actuators.

Low order controller with regional pole placement

In this paper we propose a method to find a low order controller for a single input single output linear continuous time system which guarantees that the closed loop poles are placed within some pre-specified region in the complex plane. Additionally, the method can also ensure that any subset of the closed loop poles are placed at specific pre-designed locations. Further, it is possible to ensure that the resulting controller is proper. The problem is solved by formulating it as an LMI constrained optimization problem. The proposed method is demonstrated on a power system example.

Optimization based state feedback control design for impulse elimination in descriptor systems

This article proposes an algorithm to minimize the Frobenius norm of state feedback gain matrix of a regular, linear time invariant, continuous time descriptor system. The resulting gain matrix ensures that the closed loop system is impulse-free and the associated non singular matrix is well-conditioned. By characterizing a subset of the set of non singular matrices through a linear matrix inequality, the related optimization is formulated as a semi-definite program. Numerical example illustrates the effectiveness of the proposed approach.

Reduced-order controller synthesis with regional pole constraint

ABSTRACT A reduced order output feedback controller is designed for a linear time invariant system, which guarantees that the closed-loop poles are placed within some pre-specified stability region in the complex plane. A convex approximation of the non-convex constraints is used to pose a sequence of semi-definite programs, which provide the lowest order proper controller satisfying the approximate constraints. The proposed method is demonstrated on two practical controller design applications.

Computation of state reachable points of descriptor systems

This paper considers the problem of computing the state reachable points, from the origin, of a linear constant coefficient descriptor system. A numerical algorithm is proposed that can be implemented to characterize the reachable set in a numerically stable way. The original descriptor system is transformed into a strangeness-free system within the behavioral framework followed by a projection that separates the system into its differential and algebraic parts. It is shown that the computation of the image space of two matrices, associated with the projected system, is enough to compute the reachable set (from the origin). Moreover, a characterization is presented of all the inputs by which one can reach to any arbitrary points in the reachable set. The effectiveness of the proposed approach is demonstrated through numerical examples.

Minimal state measurements for regional pole placement

The problem of minimizing the number of state measurements (and hence the number of sensors) required for placing the poles of a linear time invariant single input system with state feedback, is considered. It is assumed that only a subset of the closed loop poles are required to be placed in pre-specified locations in the complex plane. The remaining poles can assume any locations inside a pre-defined region in the complex plane. The resulting binary program with polynomial constraints is convexified using the theory of moments. Numerical examples illustrate the theory developed.

Feedback Controller Norm Optimization for Linear Time Invariant Descriptor Systems With Pole Region Constraint

An algorithm is proposed to compute a state feedback gain matrix for a linear time invariant, regular descriptor system, which ensures that i) the closed loop system is impulse-free and ii) all the finite poles are placed within a pre-defined stability region in the complex plane. The associated design flexibilities are exploited in minimizing upper bounds of the Frobenius norm of respective gain matrices. A class of linear matrix inequality (LMI) regions in the complex plane are chosen as constraints for the closed loop finite poles. By representing a subset of the nonsingular matrices through solution of an LMI, and linearizing the set of matrix inequalities, arise in the regional pole (finite poles) assignment, the associated optimizations are formulated as semidefinite programs. The effectiveness of the developed algorithm is demonstrated through numerical examples. Significant reduction in the norm of feedback gain matrix is achieved in a two generator six bus power system.