On pole assignment and optimal control theory

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.

Some points of the paper by N.K. Nichols (see ibid., vol.AC-31, p.643-5, 1986), concerning the robust pole assignment of linear multiinput systems, are clarified. It is stressed that the minimization of the condition number of the closed-loop eigenvector matrix does not necessarily lead to robustness of the pole assignment. It is shown why the computational method, which Nichols claims is robust, is in fact numerically unstable with respect to the determination of the gain matrix. In replying, Nichols presents arguments to support the choice of the conditioning of the closed-loop poles as a measure of robustness and to show that the methods of J Kautsky, N. K. Nichols and P. VanDooren (1985) are stable in the sense that they produce accurate solutions to well-conditioned problems.

This paper is concerned with the development of fully Autonomous Underwater Vehicles (AUVs) for deep ocean scientific and commercial applications. Specifically, it is regarding the design of automatic feedback control systems for AUV attitude control. The design method used in this paper formulates the AUV attitude control problem within the framework of the Optimal Control Theory. This approach determines an optimal control law that keeps the state of a dynamic system near a constant reference state while minimizing a quadratic cost functional. This study has developed four optimal control laws based on different performance criteria. First, the standard Linear Quadratic performance index was used to obtain a baseline control design. The LQR control law has been found to be undesirable because it resulted in a closed-loop system with fast and slow dynamics. In order to place the closed-loop eigenvalues close to each other, a pole placement algorithm has been used in connection with the LQR performance index. Two control laws have been designed based on this approach. Finally, a fourth design has been obtained by using a Hilbert Space Norm (also known as H/sub 2/ Norm or Root-Mean-Square) performance index. The main reason for designing a control law based on H/sub 2/ Norm performance index was to minimize the fuel consumption of the AUV for longer endurance. The stability and performance characteristics of the four optimal control laws have been analyzed via computer simulation and frequency response studies. All the closed-loop AUV systems exhibited overdamped response without any overshoot. Because the yaw and pitch channels have right hand zeros indicating the controllers have finite gain margins in these two channels. Hence, stability robustness of the AUV controllers can not be ignored and must be the subject of more detailed investigation in the future.

This paper presents a new method for the design of control laws for systems with on-off nonlinear actuators. The new methodology improves the design of control laws for pulse-modulated systems by allowing the use of continuous-time multi-inputlmultioutput design procedures. Discrete-time state feedback control gains are developed from the of continuous-time feedback gains, based on a geometric series approximation, to closely match the states of the closed-loop hybrid system to those of the original designed closed-loop continuous-time system at each sampling instant. A delay is then incorporated into a pulse-width modulator design to closely match the states of the modulated system to those of the discrete-time pulse-amplitude modulated system. The delay time is improved from previous work to provide a close match between the states for pulse durations of all sizes, in effect matching the states of the pulse-width modulated system to those of the closed-loop continuous-time system. The new control law design is applied to the problem of attitude control of the Space Shuttle Orbiter with the Hubble Space Telescope deployed on its remote manipulator. * Senior Member Technical Staff. Member A I M . t Professor, Electrical Engineering. Member AIAA. tt Aerospace Engineer. Member AIAA. Copyright O 1995 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Many paradigms are available for the synthesis of control laws for continuous-time systems, and though many methods also exist for developing discrete-time or digital controllers, it is often desirable to design a control law for the continuous-time plant. It is often more direct to determine the control specifications in the continuous-time or frequency domain. Also, direct digital control law design has difficulty in predetermining the hybrid control specifications and ignores inter sampling effects. Yet, these continuoustime control laws are generally implemented via discrete-time computer control, requiring that a discrete-time control, equivalent to the continuous-time control, be developed. The process of developing discrete-time control laws that provide closed-loop performance of the hybrid system equivalent to the performance of the designed closed-loop continuoustime system is known as redesign. Several techniques have been developed to digitally redesign continuous-time feedback controllers[l] ,[2] ,[3]. Unfortunately, the principal applied for the design of discrete-time control laws, i.e., that the amplitude of the control signal can take any finite value, which is why discrete-time control is often referred to as pulse-amplitude control (PAM), is not valid for all computer controlled systems. Systems that utilize onoff control devices, such as the reaction jet or electronic relay, can only provide a single finite value of control. To provide control with these types of actuator, either complex non-linear algorithms[4] must be developed or the control must be applied by modulating the pulse width, the pulse frequency or both the pulse width and frequency. These modulation techniques, in particular pulse-width modulation (PWM), allow the design of linear control laws to control non-linear systems. Although methods exist for the design of linear control laws for pulse width modulation systems[5], they still

In this work, we ask what the minimum data-rate is for a possibly nonlinear control law acting on a linear system to emulate the closed-loop behavior of a desired linear control law. This result gives an implicit estimate for the information flow in the feedback path of the closed-loop linear system. We formally introduce the notion of emulation and control law, and we present a data-rate theorem. Remarkably, we note that the minimum data-rate varies discontinuously with the parameters of the open and closed-loop systems. This feature is new since the usual data-rate theorem, which only requires stabilization instead of emulation, gives a continuously varying minimum data-rate.

A method is presented to place a number of system eigenvalues by means of output feedback, and to use any remaining design freedom to minimize a specified performance functional. The solution for this combination of optimal control and pole assignment is shown to be similar to that for the complete state feedback case and uses a novel pole assignment algorithm.

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. >

Adaptation fuzzy cerebellar model arithmetic controller (CMAC) neural networks are considered. Adaptation mechanisms for a fuzzy CMAC neural network are proposed to enable the construction of indirect and direct control laws. These control laws are then used to enhance the robustness of a closed-loop control system. It is shown that the fuzzy CMAC's can cope with the system's uncertainties using adaptation with no preliminary off-line learning phase being required. The adaptation laws are derived using a Lyapunov stability analysis, so that both system tracking stability and error convergence can be guaranteed in the closed-loop system. Simulation results from the two systems show a satisfactory performance of the proposed control schemes even in the presence of modelling uncertainties.

A new pole-placement algorithm for multi-input controllable systems is presented. The algorithm can be used to obtain real-valued feedback gains for arbitrarily specified real and complex, distinct or multiple eigenvalues. The multi-input controllable time-invariant system is converted to a single-input system. Pole assignment is achieved by making use of certain properties of the equation TF - AT = D , where A is the open-loop system matrix and F represents a matrix having the desired closed-loop eigenvalues.

Nonlinear Attitude Control for Large and Fast Maneuvers

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 considers control of an elastic robotic arm of two links based on nonlinear inversion and pole assignment for stabilization. The design is done in two steps. First, based on nonlinear inversion, a nonlinear controller is designed for the trajectory control of the joint angles using joint torquers. Although, the inverse controller accomplishes trajectory control of the joint angles, this excites the elastic modes of the arm. In order to damp the elastic oscillations, a stabilizer is designed for a linearized system about the terminal state using pole assignment technique. In the closed-loop system, first the inverse controller acts when a command is given, and the stabilizer automatically switches when the joint angle trajectory enters a specified neighborhood of the final commanded position.

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 .

The pole assignment problem for distributed parameter systems governed by analytic semigroups is discussed. Introducing unbounded feedback operators, we can solve the problem under a weaker restriction on the separation of the open-loop and the closed-loop poles. These types of feedback systems inherit the spectrum determined growth assumption of the open-loop systems. We first present a characteristic equation whose roots are the poles of the closed-loop system. Next we give a feedback element which realizes the desired pole assignment of the closed-loop system. We show an interesting result that an infinite number of the open-loop poles can be uniformly moved by means of a suitable linear feedback. We apply the results -to a heat system and a flexible structure for the cases of distributed controls and boundary controls.

Direct adaptive sliding mode tracking control law design method and its application to an inverted pendulum system have been presented in this paper. Firstly, problem formulation and the introduction of dynamic fuzzy logical system (DFLS) were given. Then the sliding mode control (SMC) law was constructed via DFLS which parameters were self-tuned on-line. The adaptive laws were determined so that the sliding mode satisfies reaching condition by applying the analysis of Lyapunov stability theory. Therefore, the closed-loop system is asymptotically stable and many merits can be obtained such as robustness, chattering free and adaptive characters. Finally, the presented control design was applied to an inverted pendulum system, and then the experimental results were proposed which validates the good performance of the obtained control law under various perturbations.