Pole assignment problem of a distributed parameter system with unbounded feedback

This short paper deals with the problem of pole assignment with incomplete state observation. It is shown that if the system is controllable and observable, and if n \leq r + m - 1 , an almost arbitrary set of distinct closed-loop poles is assignable by gain output feedback, where n, r , and m are the numbers of state variables, inputs and outputs, respectively. This result improves considerably the ones obtained so far about this problem. Different from the conventional approach using the characteristic equation, an approach based on the properties of the eigenspaces of the closed-loop dynamics is used in this short paper, which gives a new light on the various problems in the linear system theory. It is also shown, as a direct consequence of this result, that the minimum order of the dynamic compensator required for almost arbitrary pole assignment of overall closed-loop system is not greater than n - m - r + 1 .

Pole assignment is a well-established controller synthesis technique used in control engineering. An important drawback of the pole assignment technique is poor numerical accuracy of the resulting compensators especially for high order systems, when fixed precision arithmetic is used in calculations. Obviously there are several parameters that affect the performance of the pole assignment algorithms. It has been, for example, already shown that increasing the order of the system causes an exponential increase in the numerical errors made in pole assignment. In this paper, the effect of the ratio of real poles to all poles of the open-loop (and closed-loop) system (real pole ratio - RPR) to the performance of the pole assignment algorithms is investigated for the first time.

The basic characteristic of the transient response of a closed-loop system is determined from the closed-loop poles. In the design of closed-loop systems, it is usually necessary to adjust the open-loop poles and zeros so as to place the closed-loop poles at desirable locations in the s plane.

The pole-assignment problem of hyperbolic distributed parameter systems with boundary or pointwise inputs is studied. The problem is solved under a weaker restriction on the separation of the open-loop poles and the closed-loop ones. A characteristic equation is first given whose roots are the poles of the closed-loop system. Next it is shown that the infinite number of open-loop poles can be uniformly moved by means of a suitable linear feedback.

This paper formulates an original hierarchical self-tuning control procedure to enhance the disturbance-rejection capability of under-actuated rotary pendulum systems against exogenous disturbances. The conventional state-feedback controllers generally make a trade-off between the robustness and control effort in a closed-loop system. To combine the aforementioned characteristics into a single framework, this paper contributes to develop and augment the baseline Linear-Quadratic-Regulator (LQR) with a novel “adjustable degree-of-stability design” module. This augmentation dynamically relocates the system's closed-loop poles in the stable (left-half) region of the complex plane by dynamically adjusting a single hyper-parameter that modifies the constituents of LQR's performance index. The hyper-parameter is adaptively modulated online via a pre-calibrated hyperbolic-secant-function that is driven by state-error variables. The performance of the proposed adaptive controller is benchmarked against fixed-gain controllers via credible hardware experiments conducted on the standard QNET Rotary Pendulum setup. The experimental outcomes indicate that the proposed controller significantly enhances the system's robustness against exogenous disturbances and maintains its stability within a broad range of operating conditions, without inducing peak servo requirements.

It is shown that the problem of pole assignment by gain output feedback in linear time-invariant multivariable systems may be reduced to an algebraic optimisation problem, using the concept of virtual model following. The method requires no transformation of the open-loop system to a canonical form. A number of related problems involving combinations of pole placement with a secondary design objective are also discussed.

In this paper, we study the pole assignment problem for 3D systems. We transform the denominator of transfer functions of the closed-loop system into the product of three stable 1D polynomials, by performing 2D dynamical feedback and input transformation on the given 3D systems. Next, we consider the possibility that these 2D dynamic compensators are realizable thoroughly, and propose the counter-measure in a case that they are not realizable. We also obtain the conditions so that the closed-loop 3D systems are stable. Moreover, we calculate the dynamical dimension which is necessary for the pole assignment, and suggest a pole assignment method with the lowest dynamical dimension. >

The problem of pole assignment in linear multivariable systems by means of unity rank output feedback is studied in this paper. It is shown that when the number of specified poles is equal to the number of inputs or outputs of the system, exact pole assignment can be achieved and the feedback matrix is calculated from a set of simple linear equations. When the number of specified poles exceeds this value, the pole assignment problem becomes non-linear. Three different formulations of the problem are considered and three numerical techniques are described for the calculation of the feedback matrix. Numerical examples are given for illustration.

This paper presents a boundary controller for a flexible marine riser to suppress the riser's vibration with a top tension constraint. The flexible marine riser is described by a distributed parameter system with a partial differential equation and four ordinary differential equations. The boundary controller is designed at the top boundary of the riser based on an integral-barrier Lyapunov function to suppress the riser's tension at top. Adaptive control is designed when the system parametric uncertainty exists. With the proposed robust adaptive boundary control, uniformed boundedness under the ocean disturbance can be achieved. Stability analysis of the closed-loop system is given using the Lyapunov stability theory. Simulation results illustrate the effectiveness of the proposed boundary controller with top tension constraint.

A computational approach is proposed to solve the minimum norm and/or robust pole assignment problem for linear periodic discrete-time systems. The proposed approach uses a periodic Sylvester equation based parametrization of the periodic pole assignment problem and exploits the non-uniqueness of the problem by imposing conditions on the norm of the resulting periodic state feedback and/or on the condition numbers of the periodic eigenvector matrices of the closed-loop system. The solution method relies on using gradient search methods on suitably defined cost functions. Explicit expression of the gradients of cost functions are derived and the efficient evaluation of the cost functions and gradients is discussed. Numerical examples illustrate the effectiveness of the proposed approach.

This paper is concerned with the synthesis of constant output feedback law which possesses integrity against failures of actuator and sensor. For a linear time-invariant system, the redundancy of actuators and sensors is analyzed firstly. In the situation where actuators and sensors are redundant, the modes of open loop system which are outside a desired region of Complex plane (e.g. the open left complex plane) may remain controllable and observable after the failures of some actuator and sensor. For every such situation of permissible actuator or sensor failures, if a constant output feedback law can make the poles of close loop system in the desired region, then the feedback law is said to possess high integrity against actuator and sensor failures, A proper objective function is proposed in this paper. By using the parameter optimization method, the output feedback law possessing high integrity can be generated. A computer program is developed on the IBM PC. For a linear time invariant model of double effect evaporator and a model of a plane, the output feedback laws which possess integrity are obtained by the program. These examples illustrate the effectiveness of the method proposed in this paper. The method does not require that the open loop system is asymptotically stable, this is one of the main advantages of the method.

A new idea to control of distributed parameter systems is to induct the boundary control idea into predictive control of distributed parameter systems. Because of complexity of distributed parameter systems, control system design is the difficulty of control theory. In this paper, discrete-time boundary predictive control algorithm of first-order linear modulus-varying distributed parameter systems based on orthogonal wavelets transformation is proposed. First-order linear modulus-varying distributed parameter systems in boundary control is approximated in Haar wavelets transformation. So the predictive control proposition of distributed parameter systems has been transformed into the predictive control issue of lumped parameter systems. The boundary predictive controller is designed for the input sequence returning to the boundary predictive control rule of the original system. Simulation studies of the proposed algorithm, as well as the system robustness under uncertainty such as the parameters perturbations, and a disturb occurring to the system output are showed. The results have verified the control effectiveness of the proposed algorithm.

This paper highlights the importance of the co-efficient matrix of the numerator transfer function matrix in pole placement and stabilization assessment of multivariable control systems. The problem of pole assignment with non-unity-rank output feedback, obtained as the sum of unity rank output feedback in two stages, is discussed. The design equations of Seraji [5] formulated in terms of the appropriately partitioned co-efficient matrix of the system numerator transfer function matrix. The pole placement and stabilization assessment procedures are illustrated with numerical examples.

The non-redundant parametrization of the pole assignment problem for a n-th order system with m inputs allows to express the solution of the problem in term of n(m − 1) free parameters. These parameters can be used to fulfill additional requirements on the closed-loop system as for instance minimum norm feedback gain matrix, well conditioned eigenvector set, maximum stability radius. One of reliable numerical methods for pole assignment is the so-called Schur method. An extension of this method is proposed which computes the solution of the pole assignment problem corresponding to a non-redundant parameter set. Several possibilities are further investigated to compute minimum norm feedback matrices. An improved approach to compute minimum Frobenius-norm feedback relying on a redundant parametrization is also discussed.

This paper introduces a mathematical formula for calculating the eigenvalue differential sensitivities of the closed-loop state matrix with respect to a disturbance matrix when the closed-loop system is subjected to an additive structured perturbation. The robust pole assignment problem is then formulated as an optimization task in which the differential sensitivities of the closed-loop poles are minimized. The case of “multi-perturbations” can be considered naturally in our problem formulation. The gradient of the objective function to be minimized is then derived which permits the minimization to be proceeded effectively. Numerical examples are then employed to illustrate the design technique.