Output Feedback Pole Placement Research Articles

Output feedback pole-placement in damping region for discrete-time uncertain polytopic systems

ABSTRACT Pole placement with specified damping and decay rate is one of the primary criteria for ensuring good transient response. This control design problem has known solutions for continuous-time systems in terms of the linear matrix inequality (LMI) region. However, a systematic solution for a discrete-time system is an open problem due to the non-convexity of the constant damping locus. This paper addresses the problem of placing the closed-loop poles of linear discrete-time uncertain polytopic systems in a constant damping region. The non-convex damping region is approximated to an elliptical segment that represents an LMI region. Criteria for designing state and output feedback (static as well as dynamic) controllers are then derived. The effectiveness of the proposed approach is demonstrated with examples including a boost converter to improve its transient behaviour in the presence of input voltage and load variations.

Constrained state feedback pole placement of coupled lateral plant dynamics of RLV during the reentry phase

Stability and dynamic behaviour of a linear multivariable system mainly depend on the pole locations of the plant in the complex s plane. For general linear time-invariant multi-input multi-output (MIMO) systems, static (constant gain) output feedback pole-placement is an open nondeterministic polynomial time (NP) hard problem. This problem has attracted much attention from the control community for the last four decades. Even today, the general problem has not been solved analytically because of its highly nonlinear nature. In this paper, we have addressed the issue of finding gains without using the procedure of sequential loop closure. We have proposed an iterative numerical algorithm to solve the pole placement problem with a minimum number of feedback gains. The proposed method is used to compute the state feedback gain matrix for coupled yaw-roll dynamics of a typical reusable launch vehicle (RLV) during the reentry phase.

Design of Control Laws Based on Inverted Decoupling and Linear Matrix Inequality for a Turboprop Engine

Abstract This paper proposes a systematic approach to design control laws for a turboprop engine. The proposed approach includes interactions decoupling and control laws design based on linear matrix inequality (LMI). First, since the main objective of the turboprop engine control system is to ensure propeller-absorbed power at a constant propeller speed, the linear model of a turboprop engine can be linearized into a two-input two-output (TITO) plants, and there exist the interactions between two control loops. Because inverted decoupling can well retain the dynamic characteristics of the original system, it is used to decouple the interactions so that the TITO plant can be divided into two single-input single-output plants, that is, gas-generator shaft speed is controlled by fuel flowrate and power turbine shaft speed is controlled by blade angle. Then, the control laws are designed separately for each control loop by solving the LMI group derived from static output feedback (SOF) and regional pole placement. Finally, the proposed approach is implemented on a two-spool turboprop engine (TSTPE) integrated model. The simulation results show that there exist strong interactions between two control loops of TSTPE, applying inverted decoupling to decouple these interactions is effective, and the gas-generator shaft speed and the power turbine speed can track their commands with appropriate performance by controlling the fuel flowrate and blade angle under the action of the designed control laws and inverted decoupling.

Control Law Design for MISO System in the Presence of Actuator Position Limit Constraint

This article is about including the actuator position limit constraint while designing PD controller based on output feedback pole placement methodology for a multi-input-single-output (MISO) system. While designing control law for MISO system, designer has additional degrees of freedom which can be utilized in many ways to achieve different time and frequency domain specifications. One such goal can be to maximize the control availability at any instant of time in the presence of growing disturbance moment. In the paper it is shown that if the proportional path gains of all the loops are constrained to be in the ratio of the maximum value (actuator saturation limit) of control input of respective loops, then at steady state, total control demand will always be distributed to different control power plants in the ratio of their maximum control availability. Designing control law with the inclusion of actuator saturation limit, as constraint on proportional path gains, helps in ensuring the stability of the whole closed loop system by postponing saturation to higher disturbance values. Efficacy of the proposed method is demonstrated on a real life problem of attitude control law design for a typical launch vehicle.

Design of state and output feedback pole placement controller in the presence of slow-rate integrated measurement

Lab measurements in the industrial processes are either through instantaneous measurement at a specific time instance, known as Slow-Rate inStantaneous Measurement (SRSM), or through Slow-Rate inTegrated Measurement (SRTM), in which some amount of material is gradually collected for a period of time; so, the average of the collected material during that period is measured. In this paper, control of a system under SRTM is studied for the first time. Pole placement state feedback with feedforward controller is proposed so that both SRTM and conventional fast-rate output can track their step reference input. This controller comprises a periodically time-varying gain and an observer to estimate the slow-rate instantaneous state variable at the sampling instances. An observability condition is studied, and a modified Kalman filter is established to estimate slow-rate instantaneous state variable using SRTM. In the next step, two modified pole placement output feedback controllers are designed using SRTM. In the first control system, the aim is to track a slow-rate integrated reference input. In the second controller, fast-rate reference input is introduced to the control system. The performance of the proposed methods is evaluated through both simulation study in a pH neutralization process plant and experimental implementation on a laboratory scale Quadruple Tank pilot plant.

The static output feedback from the invariant point of view

This paper addresses the static output feedback (SOF) pole placement problem of linear minimal systems with |$m$| inputs, |$r$| outputs, |$n$| states and non-zero controllability and observability indices, as an eigenstructure assignment problem. A mosaic Hankel matrix having SOF-invariant properties is used to reformulate the coupled Sylvester equations. Necessary conditions for assignability and stabilizability as well as sufficient for linearity are presented together with an algorithm calculating the solution.

Application of Newton's Method with Analytical Computation of Jacobian Matrix to Solve Multi-linear Pole Assignment Equations

Abstract Output feedback pole placement problem is not solvable analytically for the plants having either number of inputs or number of outputs, more than two. There is even lack of a good numerical method which solves any general pole assignment problem. Due to the multi-linear nature of the pole placement problem there is a possibility to utilize multi-linear structure and arrive at a better numerical solution. This paper shows that it is possible to compute analytically the Jacobian and Hessian matrix in an easy manner and utilize them in an iterative numerical method to solve the pole-placement problem. Newton-Raphson method is used with analytical solution of Jacobian matrix to give an iterative solution of pole placement equations with better percentage success rate than the other methods quoted in literature.

Semi Analytical Solution of Constrained Output Feedback Pole Placement in Multi-Input Systems

Pole-placement is one of the oldest control design method used to modify the behavior of a given plant dynamics. In the literature, analytical solution of output feedback pole placement is available only for two-input two-output systems. For general linear time-invariant, multi-input, multi-output (MIMO) systems, static (constant gain) output feedback pole placement is an open non-deterministic polynomial-time (NP) hard problem. This paper proposes a semi-analytical solution of constrained (minimum number of nonzero gains) output feedback pole placement problem in MIMO systems. To get the semi-analytical solution, first an analytical formula is developed for pole-placement in single-input, multi-output (SIMO) systems. This formula is an extension of Bass-Gura's formula used for state feedback pole placement in single-input systems. Along with the gains needed for partial pole-placement, this novel formula computes the coefficients of characteristic polynomial corresponding to the remaining poles of the plant. Using this analytical formula a novel iterative procedure is developed to place all the poles of a multi-input system using output feedback. Proposed iterative method needs only the solution of linear equations at each step of iteration. Starting from the known initial guess, it searches for the solution in a lower dimension space compared to other numerical methods. The algorithm computes a full rank feedback gain matrix. For a decoupled m-input system the proposed method computes the output feedback gains in exactly m-steps, non-iteratively.

Output feedback pole placement for linear time-varying systems with application to the control of nonlinear systems

The output feedback pole placement problem is solved in an input–output algebraic formalism for linear time-varying (LTV) systems. The recent extensions of the notions of transfer matrices and poles of the system to the case of LTV systems are exploited here to provide constructive solutions based, as in the linear time-invariant (LTI) case, on the solutions of diophantine equations. Also, differences with the results known in the LTI case are pointed out, especially concerning the possibilities to assign specific dynamics to the closed-loop system and the conditions for tracking and disturbance rejection. This approach is applied to the control of nonlinear systems by linearization around a given trajectory. Several examples are treated in detail to show the computation and implementation issues.

CDM-based design and performance evaluation of a robust AQM method for dynamic TCP/AQM networks

A new robust AQM strategy for dynamically varying TCP/AQM networks is proposed and its performance is investigated through computer simulations in MATLAB and ns-2 environments. The developed AQM is designed based on coefficient diagram method (CDM), which is a new indirect pole placement method that considers the speed, stability and robustness of the closed loop system in terms of time domain specifications. Simulation results indicate that the new method (CDM-AQM) performs very well for network variations both in topology and traffic. Besides, a new adaptive controller based on CDM as an AQM method is introduced. In the developed adaptive AQM (ACDM), the output feedback pole placement is implemented in an indirect model reference adaptive control (MRAC) scheme, in which the closed loop characteristic polynomial is determined by CDM and the system parameters are estimated using a modified recursive gradient method. This method preserves the existing features of CDM controller in the control of networks with unknown or time-varying parameters. Simulation results illustrate high capability of the controller in coping with the network variations.

Generalized pole placement via static output feedback: A methodology based on projections

This paper presents an algorithm for solving static output feedback pole placement problems of the following rather general form: given n subsets of the complex plane, find a static output feedback that places in each of these subsets a pole of the closed-loop system. The algorithm presented is iterative in nature and is based on alternating projection ideas. Each iteration of the algorithm involves a Schur matrix decomposition, a standard least-squares problem and a combinatorial least-squares problem. While the algorithm is not guaranteed to always find a solution, computational results are presented demonstrating the effectiveness of the algorithm.

POLE PLACEMENT VIA OUTPUT FEEDBACK: A METHODOLOGY BASED ON PROJECTIONS

This paper presents an algorithm for solving static output feedback pole placement problems of the following rather general form: given n subsets of the complex plane, find a static output feedback that places in each of these subsets a pole of the closed loop system. The algorithm presented is iterative in nature and is based on alternating projection ideas. Each iteration of the algorithm involves a Schur matrix decomposition, a standard least squares problem and a combinatorial least squares problem. Computational results are presented demonstrating the effectiveness of the algorithm.

Pole placement controllers for linear time-delay systems with commensurate point delays

This paper investigates the exact and approximate spectrum assignment properties associated with realizable output-feedback pole-placement type controllers for single-input single-output linear time-invariant time-delay systems with commensurate point delays. The controller synthesis problem is discussed through the solvability of a set of coupled diophantine equations of polynomials. An extra complexity is incorporated to the above design to cancel extra unsuitable dynamics being generated when solving the above diophantine equations. Thus, the complete controller tracks any arbitrary prefixed (either finite or delay-dependent) closed-loop spectrum. However, if the controller is simplified by deleting the above mentioned extra complexity, then the robust stability and approximated spectrum assignment are still achievable for a certain sufficiently small amount of delayed dynamics. Finally, the approximate spectrum assignment and robust stability problems are revisited under plant disturbances if the nominal controller is main-tained. In the current approach, the finite spectrum assignment is only considered as a particular case to the designer's choice of a (delay-dependent) arbitrary spectrum assignment objective.

Compensator based output feedback sliding mode controller design

Recent work has shown that the problem of hyperplane design for output feedback sliding mode controllers is equivalent to a static output feedback pole placement problem for a certain subsystem. It is well known that the dimensionality requirements for this type of problem can be obviated by using a dynamic compensator. In this paper, a new parameterization of both the hyperplane and the compensator is proposed. The advantages of this approach are twofold: firstly it becomes straightforward to construct numerical algorithms for dynamic output feedback sliding mode hyperplane design, and secondly it enables an intuitive procedure to be employed for the design of the controller. The resulting control strategy uses only the plant outputs and the compensator states, yet is shown to guarantee the existence of a sliding motion.

Output feedback pole placement with dynamic compensators

Derives a new rank condition which guarantees the arbitrary pole assignability of a given system by dynamic compensators of degree at most q. By using this rank condition the authors establish several new sufficiency conditions which ensure the arbitrary pole assignability of a generic system. The authors' proofs also come with a concrete numerical procedure to construct a particular compensator which assigns a given set of closed-loop poles.

On Output Feedback via Grassmannians

The output feedback pole placement map $\chi $ has been studied by using the central projection model. Necessary and sufficient conditions for $\chi $ being dominant are given. $2 \times 2$ systems of McMillan degree 4 and $2 \times 3$ systems of McMillan degree 6 are studied intensively.

Output feedback pole placement in the design of suboptimal linear quadratic regulators†

A design oriented methodology is presented for the construction of low-order, suboptimal, output feedback compensators. The design is suboptimal in the sense that it retains an l-dimensional eigenspace from a reference optimal state-feedback linear quadratic regulator problem. If r is the number of outputs, then for fixed l > r, it is known that the design requires a dynamic compensator of dimension l — r. The parameters of the compensator are determined by the solution of an associated output feedback pole placement problem. Using iterative, dyadic pole placement, a procedure is given which determines the solution of this problem, without a priori assumptions on the dimension of the compensator. The design is also extended to the class of stabilizable systems, and the resulting compensator is shown to possess a separation property.

Correction to "Some new bounds related to output feedback pole placement"

Correction to "Some new bounds related to output feedback pole placement"

Output feedback decoupling and pole placement in linear time-invariant systems

The question of closed-loop pole placement for linear multivariable systems decoupled by output feedback is considered. The class of decoupling feedback matrices is extended to include those which are proper functions of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> . This result insures that all uncancelled poles of the resulting closed-loop system can be reassigned to any desired location by suitable choice of the feedback matrix.

Some new bounds related to output feedback pole placement

A number of new results regarding linear output feedback compensation are presented. In particular, it is shown that the rank of an appropriately defined real matrix Ω represents an upper bound on the number of closed-loop poles which can be completely and arbitrarily assigned via constant gain output feedback. A new bound on the minimum number of dynamical elements required for complete and arbitrary closed-loop pole placement is also defined in terms of the observability index of a certain single-input system.