Partial Pole Placement Research Articles

Partial pole placement of H∞ based PSS design using numerator-denominator perturbation representation

A new H∞ PSS design method which uses the numerator-denominator perturbation representation and includes the partial pole placement technique and a new weighting function selection method is proposed. This overcomes certain conventional H∞ PSS design algorithm limitations. A sixth-order machine model is used to increase the accuracy of selected weighting functions. A robust PSS has been successfully designed for single and two-machine systems by treating the highly nonlinear characteristic of the power system as model ‘uncertainty’. The design is verified to have better performance for a wide range of system operating conditions when compared with the conventional PSS designs.

Observer-Based Quadratic Stabilization with Dominant Pole Placement

This paper proposes an observer-based output feedback design method which attains both quadratic stabilization and dominant pole placement. For a system whose state signal is unavailable and whose coefficient matrices have structured uncertainty, the method gives a controller which satisfies an H∞ norm condition for quadratic stability and the partial pole configuration, under a certain sufficient condition. This is achieved by means of the parameterization of partial pole placement which has already been known for the state feedback case. Degradation due to adopting state observers is then discussed. A numerical example is given to illustrate the proposed method.

Partial Pole Placement with Specified Stability Margins

This paper proposes a design method of a state feedback gain which simultaneously achieves specified gain and phase margins and partial pole placement. Gain and phase margins are guaranteed by imposing quadratic constraints on open-loop transfer functions. Also, partial pole placement is done by using the same gain parametrization as in the polynomial ILQ method. It is shown that our design problem is formulated as a feasibility problem with LMI constraints and a non-linear equality constraint.

A new H∞ based PSS design using numerator–denominator perturbation representation

Abstract A new H∞ PSS design method which uses the numerator–denominator perturbation representation and includes the partial pole placement technique and a new weighting function selection method is proposed. This overcomes certain conventional H∞ PSS design algorithm limitations. A robust PSS has been successfully designed by treating the highly non-linear characteristic of the power system as model ‘uncertainty’. The design is verified to have better performance for a wide range of system operating conditions when compared with the conventional PSS design.

Partial pole placement by LQ regulators: an inverse problem approach

This paper gives a necessary and sufficient condition under which a state feedback control law places part of the closed-loop poles exactly at specified points and, at the same time, is linear quadratic optimal for some quadratic weightings. This is made possible by means of a solution to the inverse problem of optimal control. A design example is given to illustrate the result.

Robust digital control using pole placement with sensitivity function shaping method

SUMMARY It is shown in this paper that based on identified discrete-time models a robust digital linear controller can be designed using the pole placement method combined with sensitivity function shaping in the frequency domain. Two di⁄erent design techniques are presented. The first one is based on the shaping of the sensitivity functions using the fixed parts in the controller and the auxiliary poles of the closed loop while keeping the dominant poles in the desired places. The main idea of the second one is to determine a weighting filter for the output sensitivity function in an H = optimization approach which assures partial pole placement and desired performances. In this technique the weighting filter is interpreted as the inverse of the desired output sensitivity function and is computed using a constrained optimization program. The application of this technique on a flexible transmission system is presented. ( 1998 John Wiley & Sons, Ltd. The controller design methodology considered in this paper is based on pole placement combined with the shaping of the sensitivity function. The computation of the controller in the pole placement technique requires the specification of the desired closed-loop poles (the nominal stability problem) and of some fixed parts of the controller for the rejection of disturbances at various frequencies (the nominal performance problem). However, this is not enough to guarantee the robustness of the design with respect to the plant model uncertainties (the robust stability and robust performance problem). A robust controller design requires also the shaping of the sensitivity functions. The sensitivity functions, particularly the output sensitivity function, are key indicators for the nominal and robust performance as well as for the robust stability of the closed-loop system. The inverse of the maximum value of the output sensitivity function, i.e., the inverse of its H = norm, gives the minimum distance between the Nyquist plot of the open-loop system and the critical point [!1, j0]. This quantity, called the modulus margin, is a much more significant robustness indicator than the phase and gain margins. On the other hand, conditions for assuring a certain delay margin which is also a very important robustness indicator, particularly in the high frequency region, can also be expressed in terms of the shape of the output sensitivity function. It seems therefore reasonable to combine the pole placement with the shaping of the output

Improving the robustness of H/sub ∞/-PSSs using the polynomial approach

This paper proposes a new design procedure of H/sub /spl infin//-PSS (H/sub /spl infin//-optimal PSS) by incorporating high-frequency roll-off into the design using improper weighting function. This design is based on the polynomial approach where the 'numerator-denominator' uncertainty representation is used for practical uncertainty representations. Furthermore, to improve the dynamic and transient stabilities of the control system, the partial pole placement technique is employed to reassign dominant poles at suitable locations in the left-half s-plane. The effectiveness of using /spl Delta//spl omega/ (shaft speed) input and /spl Delta/P/sub e/ (electrical power) input H/sub /spl infin//-PSSs to damp local oscillation modes is also investigated by comparing the performances of these H/sub /spl infin//-PSSs. Simulation results show that the proposed approach is effective in improving the performance and robustness of the system.

Robust partial pole‐placement via gradient flow

This paper provides a computational procedure for the robust partial pole-placement problem. The algorithm is based on a gradient flow formulation on a differentiable potential function which provides a minimizing solution for the Frobenius condition number of the closed-loop state matrix. The algorithm faces no singularity problem with the resulting eigenvector matrix. Convergence properties of the algorithm are discussed. A numerical example is employed to illustrate the technique and comparison to other existing methods is made. © 1977 John Wiley & Sons, Ltd.

Robust partial pole-placement via gradient flow

This paper provides a computational procedure for the robust partial pole-placement problem. The algorithm is based on a gradient flow formulation on a differentiable potential function which provides a minimizing solution for the Frobenius condition number of the closed-loop state matrix. The algorithm faces no singularity problem with the resulting eigenvector matrix. Convergence properties of the algorithm are discussed. A numerical example is employed to illustrate the technique and comparison to other existing methods is made. © 1977 John Wiley & Sons, Ltd.

Partial Pole Placement by Discrete-Time LQ Control and its Sampled-Data Performance

This paper gives a design method for discrete-time optimal control, and then studies its performance from a sampled-data point of view. First, as discrete-time control, a state feedback gain is designed which places part of the closed-loop poles exactly at specified points inside the unit circle, and is linear quadratic (LQ) optimal for some weightings at the same time. This is achieved by placing the rest of the poles sufficiently close to the origin, thereby satisfying a modified circle criterion, a solution to the inverse problem of discrete-time LQ control. Secondly, it is checked whether thus obtained (pure) discrete-time feedback is again optimal as a sampled-data control system; i.e., whether it minimizes some continuous-time performance index.

AN IMPROVED H∞ POWER SYSTEM STABILIZER

In designing power system stabilizer (PSS), a major difficulty is to successfully treat the system's uncertainties. These uncertainties arise because of changes in operating conditions, approximations in modeling, parameter variations caused by faults, etc. It is known that in the presence of uncertainties, conventional methodologies such as linear optimal technique, adaptive controls, etc., may fail to guarantee the stability of the system. On the other hand, the H∞ control theory provides potential ability to overcome this problem. However, some limitations still exist in the treatment of uncertainty. That is, the existing standard H∞-PSSs cannot adequately treat the system uncertainties. Moreover, performance problems can arise in this approach due to the pole-zero cancellation phenomenon. To deal with the above mentioned limitations, a new design methodology for H∞-PSS based on the 'numerator-denominator' uncertainty representation is proposed, where partial pole placement technique is used. Simulation results suggest that the proposed PSS is more robust and less sensitive to disturbances than both the conventional PSS and the standard H∞-PSS. The superiority of the proposed PSS has also been confirmed by computational results of stability margin as well as the Bode plots of the sensitivity functions.

Discrete-time pole placement with stable controller

The classical polynomial approach to pole placement involves the computation of a particular solution of a given Diophantine equation. The solution corresponds to a proper controller whose order is fixed by the number of nonminimum phase zeros of the plant transfer function, by its order, and by the desired closed-loop characteristic polynomial. This controller might be unstable for some desired closed-loop characteristic polynomials, even though the plant can be stabilized by a stable controller. Now, unstable controllers are known to deteriorate closed-loop system performances, compared with stable ones. Hence, we propose a procedure to build, for a specific class of plants, a stable pole placement controller when the classical pole placement controller is unstable. The properties of both stable and unstable controllers are compared via simulations. It appears that, for the considered plants, the stable controller improves at least one of the following properties, compared with its unstable counterpart: step response transient, gain margin, measurement noise rejection. The results obtained with our procedure are also compared with the results obtained via Middleton's approach, which consists in modifying the desired closed-loop poles so as to obtain a stable and minimum phase controller with the classical pole placement method. Finally, our pole placement design with stable controller is applied to solve a particular simultaneous partial pole placement problem.

Pole-zero cancellations and closed-loop properties of an H∞ mixed sensitivity design problem

Abstract We examine an H ∞ mixed sensitivity design problem in order to gain an understanding of the pole-zero cancellation phenomenon that is known to occur with such designs. It is shown how the pole-zero cancellation phenomenon is dependent upon the choice of weighting functions used in the problem formulation, and a particular construction of weighting function is given that gives the designer freedom to exploit or prevent the phenomenon. Furthermore, this weighting function is then a mechanism for (partial) pole placement. As a consequence of the construction of this weighting function, it is shown that for a certain class of plant, the Normalized Coprime Factor design procedure of Glover and McFarlane (1989, Robust Controller Design Using Normalized Coprime Factor Plant Descriptions . Springer-Verlag, Berlin) gives rise to (possibly undesirable) pole-zero cancellations. The results are presented for multivariable systems, and are illustrated with a simple design example.

Microcomputer-based robust digital control for the glucoregulatory system

The regulaton of glucose in a metabolic system was achieved based on a fourth-order well-identified digital model and a pre-determined digital control. Assuming the model is sufficiently accurate to avoid the need for measurement of the glucose concentration, an open-loop digital controller is designed. A partial pole placement technique together with the dominant roots concept are used to synthesize the discrete time controller. The parameters of the controller are determined from the solution of a set of linear simultaneous equations. The closed-loop dynamics are dictated by the type of the dominant poles. A set of well-behaved responses and internal states were obtained with variable overshoot. The dynamics have shown to be comparatively fast and the controller was shown to be robust using the bounding polynomials suggested by Kharitonov’s stability criterion.

Projection and deflation method for partial pole assignment in linear state feedback

Two projection methods are proposed for partial pole placement in linear control systems. These methods are of interest when the system is very large and only a few of its poles must be assigned. The first method is based on computing an orthonormal basis of the left invariant subspace associated with the eigenvalues to be assigned and then solving a small inverse eigenvalue problem resulting from projecting the initial problem into that subspace. The second method can be regarded as a variant of the Weilandt deflation technique used in eigenvalue methods.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Robustness in partial pole placement

The robustness of state feedback solutions to the problem of partial pole placement obtained by a new projection procedure is examined. The projection procedure gives a reduced-order pole assignment problem. It is shown that the sensitivities of the assigned poles in the complete closed-loop system are bounded in terms of the sensitivities of the assigned reduced-order poles, and the sensitivities of the unaltered poles are bounded in terms of the sensitivities of the corresponding open-loop poles. If the assigned poles are well-separated from the unaltered poles, these bounds are expected to be tight. The projection procedure is described in [3], and techniques for finding robust (or insensitive) solutions to the reduced-order problem are given in [1], [2].

Transcendental and Interpolation Methods in Simultaneous Stabilization and Simultaneous Partial Pole Placement Problems

The existence of a compensator which simultaneously renders a given r-tuple of multiinput-multioutput p x m linear dynamical systems internally stable is investigated. In particular, a set of simultaneously stabilizable r-tuples of plants is parametrized, and it is shown that, provided r = max(m,p) or less, the above set is semialgebraic and dense in the space Sigma of r-tuples of plants. An extension of the classical pole placement and stabilization problems is considered, and the simultaneous partial pole placement problem is investigated.

Simultaneous partial pole placement: A new approach to multimode system design

Simultaneous partial pole placement of a family of single-input single-output plants is proposed as a generalization of the classical pole placement and stabilization problems. This problem finds application in the design of a compensator for a family of linear dynamical systems. In this note we show that the proposed problem is equivalent to a new class of transcendental problem using stable, minimum phase rational functions with real coefficients. A necessary condition for the solvability of the associated transcendental problem is obtained. Finally, a counterexample to the following conjecture is obtained-pairs of simultaneously stabilizable plants of bounded McMillan degree have simultaneously stabilizing compensators of bounded McMillan degree.

Parameter insensitive pole assignment for scalar input systems

Under the assumption that the parameters affect the systems linearly, arbitrary partial insensitive pole placement problem for scalar input systems have been solved. A new definition for partial pole placement, which takes into account the effect of the unknown varying parameters in the system, has been used. The result is first given for a system with one parameter, and then, it is extended to systems with more than one parameter. The dual of the same result is also given. The paper also emphasizes the structural properties of systems with varying parameters.