Partial Pole Placement Research Articles

Enhancing disturbance rejection in boost converter for CPL: A controller design approach with partial pole placement and approximate model matching

In DC microgrids, the cascade operation of the DC-DC converters leads to a constant power load fed by the boost converter, resulting in unstable behavior in addition to the non-minimum phase dynamics. An enhanced load disturbance rejection control scheme based on partial pole placement and an approximate model matching technique is proposed. This approach is inspired by the well-known direct synthesis method to achieve optimal load disturbance rejection performance. The effectiveness of the proposed control scheme has been validated through hardware implementation, demonstrating its capability in setpoint tracking and load disturbance rejection under variations in load power and input voltage. The robust stability of the suggested scheme has been investigated through the Kharitonov theorem and Monte-Carlo simulation, considering different performance matrices. The suitability of the presented scheme has been established by analyzing the comparative performance of recently reported works. The proposed controller reduces peak overshoot and settling time by ∼50%, handles system uncertainties up to 75%, and outperforms FOPID controllers tuned with particle swarm optimization, queen bee genetic algorithm, and chaos game optimization methods.

State Observing Method and Active Vibration Control Based on Virtual Sensing

State feedback is mostly used in the active vibration control of structural systems. However, the most challenging problem is to achieve the whole state of the structure. Another issue in active vibration control is that, in many cases, only some particular modes need to be controlled, while the other modes can remain unchanged in order to save on the costs of the control. To solve the above issues, this paper proposes a state-observing method and conducts a non-spillover partial pole placement control using a multistep method. Using the finite element method, the electromechanical coupling equations of motion of the beam are formulated. The reduced finite element model is then obtained by using the system equivalent reduction expansion process method. Introducing the real measurements by sensors and applying the local corresponding principle, the modes of the structural system are corrected, which are then utilized to predict all the degrees of freedom without measuring by sensors. Then, the non-spillover partial pole placement based on the state feedback is conducted using the multistep method. The optimal locations of actuators are derived by the genetic algorithm to control the specific structural modes effectively. The numerical simulations and experimental studies are also carried out.

Insights on equidistributed real spectral values in second-order delay systems: Perspectives in partial pole placement

In this work, we show that the coexistence of the maximal number of real spectral values of generic single-delay retarded second-order differential equations guarantees the realness of the rightmost spectral value. From a control theory standpoint, this entails that a delayed proportional-derivative (PD) controller can stabilize a delayed second-order differential equation. By assigning the maximum number of negative roots to the corresponding characteristic function (a quasipolynomial), we establish the conditions for asymptotic stability. If the assigned real spectral values are uniformly distributed, we specify a necessary and sufficient condition for the rightmost root to be negative, thus guaranteeing the exponential decay rate of the system’s solutions. We illustrate the proposed design methodology in the delayed PD control of the damped harmonic oscillator. It is worth mentioning that this work represents a natural continuation of Amrane et al. (2018) and Bedouhene et al. (2020), addressing the problem of coexisting real spectral values for linear dynamical systems including delays in their models.

Prescribing transport equation solution's decay via multiplicity manifold and autoregressive boundary control

AbstractThis article addresses the boundary control problem of the transport equation. Namely, we propose a control method, which is merely a delayed output feedback relying on a partial pole placement idea, that consists in assigning an appropriate exponential decay rate to the closed‐loop system's solution. The proposed control structure appearing in the transport boundary, which has proven its effectiveness in controlling finite dimensional systems, consists of an autoregressive relation linking the transport equation's input and output. The obtained result provides an analytical lower bound for the solution's exponential decay. The contribution is concluded with a numeric discussion on the concrete implementation of the method with a special emphasis on the delay seen as a control parameter.

On the pole placement of scalar linear delay systems with two delays

AbstractThis paper concerns some spectral properties of the scalar dynamical system defined by a linear delay-differential equation with two positive delays. More precisely, the existing links between the delays and the maximal multiplicity of the characteristic roots are explored, as well as the dominance of such roots compared with the spectrum localization. As a by-product of the analysis, the pole placement issue is revisited with more emphasis on the role of the delays as control parameters in defining a partial pole placement guaranteeing the closed-loop stability with an appropriate decay rate of the corresponding dynamical system.

Eigenvalue sensitivity minimisation for robust pole placement by the receptance method

The problem of robust pole placement in active structural vibration control by the method of receptance is considered in this paper. Expressions are derived for the eigenvalue sensitivities to parametric perturbations, which are subsequently minimised to improve performance robustness of the control of a dynamical system. The described approach has application to a vibrating system where variations are present due to manufacturing and material tolerances, damages and environment variabilities. The closed-loop eigenvalue sensitivities are expressed as a linear function of the velocity and displacement feedback gains, allowing their minimisation with carefully calculated feedback gains. The proposed algorithm involves curve fitting perturbed frequency response functions, FRFs, using the rational fraction polynomial method and implementation of a polynomial fit to the individual estimated rational fraction coefficients. This allows the eigenvalue sensitivity to be obtained entirely from structural FRFs, which is consistent with the receptance method. This avoids the need to evaluate the M,C,K matrices which are typically obtained through finite element modelling, that produces modelling uncertainty. It is also demonstrated that the sensitivity minimisation technique can work in conjunction with the pole placement and partial pole placement technique using the receptance method. To illustrate the working of the proposed algorithm, the controller is first implemented numerically and then experimentally.

New Features of P3δ Software. Insights and Demos

This paper presents the software entitled “Partial Pole Placement via Delay Action,” or “P3δ” for short. P3δ is a Python software with a friendly user interface for the design of parametric stabilizing feedback laws with time-delays for dynamical systems. After recalling the theoretical foundation of the so-called “Partial Pole Placement” methodology we propose as well the main features of the current version of P3δ. We illustrate its use in feedback stabilization of several control systems operating under time delays.

Over one Century of Spectrum Analysis in Delay Systems: An Overview and New Trends in Pole Placement Methods

Pole placement represents a classical method for controlling finite-dimensional linear time-invariant systems, largely covered in the open literature. Basically, it consists of placing the poles of the closed-loop system in some predetermined loci in the complex plane. This paper discusses some of the extensions of this method to linear systems described by delay-differential equations. Among others, the finite spectrum assignment (FSA), the continuous pole placement (CPP) and the partial pole placement (PPP) approaches are presented and illustrated through some simple low-order dynamical systems.

Observer Design in LTI Time-Delay Systems using Partial Pole Placement with Applications

In order to estimate the state of a dynamical system, we address the problem of designing an observer for linear time-invariant (LTI) dynamical systems including a time-delay. Sufficient conditions for the existence of the proposed observer are given using partial placement of the poles for the error. Namely, we exploit the multiplicity-induced-dominance property of the characteristic function corresponding to the system's error. The effectiveness of the proposed observer design is shown in both state lag and input lag respectively through the problems of Mach number control in a wind tunnel and stabilization of the inverted pendulum on a cart by using the delay.

Padé Approximation and Hypergeometric Functions: A Missing Link with the Spectrum of Delay-Differential Equations

It is well known that rational approximation theory involves degenerate hyperge-ometric functions and, in particular, the Padé approximation of the exponential function is closely related to Kummer hypergeometric functions. Recently, in the context of the study of the exponential stability of the trivial solution of delay-differential equations, a new link between the degenerate hypergeometric function and the zeros distribution of the characteristic function associated with linear delay-differential equations was emphasized. Such a link allowed the characterization of a property of time-delay systems known as multiplicity-induced-dominancy (MID), which opened a new direction in designing low-complexity controllers for time-delay systems by using a partial pole placement idea. Thanks to their relations to hypergeometric functions, we explore in this paper links between the spectrum of delay-differential equations and Padé approximations of the exponential function. This note exploits and further comments recent results from [I. Boussaada, G. Mazanti and S-I. Niculescu. 2022, Comptes Rendus. Mathématique] and [I. Boussaada, G. Mazanti and S-I. Niculescu. 2022, Bulletin des Sciences Mathématiques].

Design of Quasipolynomial-Based Controllers with Dynamical Parameters - Application to Active Vibration Damping

In order to achieve a partial pole placement for linear time-invariant systems including time-delays in their models’ representation, a method for the design of quasipolynomial-based controllers has been proposed in recent works. The ensuing controllers correspond to some output feedback control laws with constant parameters. It appears that such a controller has a limited number of degrees of freedom limiting the potential performances in closed-loop. To overcome this issue, we propose to modify the previous quasipolynomial-based controller by using dynamical parameters in their design. It turns out that the use of dynamical parameters corresponds to linear filtered terms in the control law of the original one. Such a controller is applied to the active vibration damping problem for a piezo-actuated flexible structure.

LMI-based design of state-feedback controllers for pole clustering of LPV systems in a union of 𝒟 R -regions

This paper introduces an approach for the design of a state-feedback controller that achieves pole clustering in a union of -regions for linear parameter varying systems. The design conditions, obtained using a partial pole placement theorem, are eventually expressed in terms of linear matrix inequalities. In addition, it is shown that the approach can be modified in a shifting sense. Hence, the controller gain is computed such that different values of the varying parameters imply different regions of the complex plane where the closed-loop poles are situated. This approach enables the online modification of the closed-loop performance. The effectiveness of the proposed method is demonstrated by means of simulations.

New Features of P3δ software: Partial Pole Placement via Delay Action

This paper presents the software Partial Pole Placement via Delay Action, or P3$\delta$ for short. P3$\delta$ is a Python software with a friendly user interface for the design of parametric stabilizing feedback laws with time-delays, thanks to two properties of the distribution of quasipolynomials' zeros, called multiplicity-induced-dominancy and coexisting real roots-induced-dominancy. After recalling recent theoretical results on these properties and their use for the feedback stabilization of control systems operating under time delays, the paper presents the main features of the current version of P3$\delta$. We detail, in particular, the assignable admissible region (the set of allowable dominant roots and the corresponding delay), which helps the user in the choice of input information, allowing a reliable stabilizing delayed feedback. We also present the newly set online version of P3$\delta$.

Active vibration control of axisymmetric membrane through partial pole placement

Abstract This paper deals with the problem of active vibration control for a flexible axisymmetric membrane. This mechanical system is equipped with two piezoelectric circular chips where one of them works as an actuator, whereas the other is used as a sensor. Both are glued on the membrane, one on each side, and centered according to its axis of symmetry. The model of this system is obtained from a finite element modelling, leading to a linear state-space model. The design of the proposed controller is based on delayed proportional actions. We exploit a property called Coexisting Real Roots inducing Dominancy to an assignment of spectral values in an appropriate sector corresponding to a desired damping. The aim of this work is to examine the performances of the proposed output feedback controller in terms of vibration damping of the main observable and controllable vibrating modes.

Stability, Delays and Multiple Characteristic Roots in Dynamical Systems: A Guided Tour

This paper presents a guided tour of some specific problems encountered in the stability analysis of linear dynamical systems including delays in their systems' representation. More precisely, we will address the characterization of multiple roots of the corresponding characteristic function with a particular emphasis on the way these roots are affected by the system's parameters and the way that they can be used to control. The paper covers several approaches (perturbation techniques, hypergeometric functions) leading to some methods and criteria (frequency-sweeping, multiplicity-induced-dominancy) that can be implemented (software toolboxes) for analyzing the qualitative and quantitative properties induced by the delays and other parameters on the system's dynamics. A particular attention will be paid to the so-called partial pole placement method based on the multiplicity-induced-dominancy property. The presentation is as simple as possible, focusing more on the main intuitive ideas and appropriate mathematical reasoning by analogy in the presentation of the theoretical results as well as their potential use in practical applications. Illustrative examples complete the paper.

Pole Assignment for Active Vibration Control of Linear Vibrating Systems through Linear Matrix Inequalities

This paper proposes a novel method for pole placement in linear vibrating systems through state feedback and rank-one control. Rather than assigning all the poles to the desired locations of the complex plane, the proposed method exactly assigns just the dominant poles, while the remaining ones are free to assume arbitrary positions within a pre-specified region in the complex plane. Therefore, the method can be referred to as “regional pole placement”. A two-stage approach is proposed to accomplish both the tasks. In the first stage, the subset of dominant poles is assigned to exact locations by exploiting the receptance method, formulated for either symmetric or asymmetric systems. Then, in the second stage, a first-order model formulated with a reduced state, together with the theory of Linear Matrix Inequalities, are exploited to cluster the subset of the unassigned poles into some stable regions of the complex plane while keeping unchanged the poles assigned in the first stage. The additional degrees of freedom in the choice of the gains, i.e., the non-uniqueness of the solution, is exploited through a semidefinite programming problem to reduce the control gains. The method is validated by means of four meaningful and challenging test-cases, also borrowed from the literature. The results are also compared with those of classic partial pole placement, to show the benefits and the effectiveness of the proposed approach.

The minimum norm multi-input multi-output receptance method for partial pole placement

Abstract A closed-form analytical solution is developed for the first time that fully addresses the problem of choosing feedback gains that minimize the control effort required for partial pole placement in multi-input, multi-output systems. The norm of the feedback gain matrix is shown to take the form of an inverse Rayleigh quotient, such that the optimal closed-loop system eigenvectors are given as a function of the dominant (highest) eigenvectors of the matrix in the quotient. The feedback gains that deliver the required pole placement with minimum effort may then be determined using standard procedures. The original formulation by the receptance method proposed an arbitrary choice of the closed loop eigenvectors that assigned the poles exactly but was generally wasteful of control effort that might otherwise be conserved or put to good use in satisfying additional control objectives. The analytical solution is validated against a set of numerical examples.

Damped Nyquist Plot for a pole placement design of power system stabilizers

This paper proposes a new frequency domain control design method through the use of a modified Nyquist diagram with an embedded partial pole-placement capability. The proposed Damped Nyquist Plot (DNP) method evaluates the open loop transfer function (OLTF) along a line of constant damping ratio to provide a graphical tool to design power system stabilizers (PSS). The graphical tool shows how the closed loop poles move around this damping ratio line for different choices of PSS parameters, all of them placing a selected pair of complex-conjugate poles at the same desired point in the complex plane. New formulas are developed to determine the exact PSS parameters that promote the several possibilities of partial pole placement considering the parameter ranges used in tuning practice. These formulas can also be used for phase compensation of conventional methods of PSS design, such as GEP or Residue Angle Compensation. Multiple PSSs can be tuned using the DNP method sequentially. The proposed method is applied to two stabilization examples, the first of tutorial nature and the second is the large actual Brazilian Interconnected Power System.

Sensor fusions for constant thrust aircraft navigation in pitch plane

Constant thrust aircraft navigation is an altitude hold mode in air transportation systems, air refueling systems, traffic alert and collision avoidance system and automatic dependent surveillance broadcast based collision avoidance, as well as in surveillance operations of unmanned aerial vehicles. In pitch plane, determination of the elevator input options available to maneuver the aircraft from its altitude hold mode using constant thrust is an important problem. Such inputs are useful to avoid time critical obstacles or collisions sensed in the horizontal plane. In this paper, it is shown that sensor fusions in feedback path are particularly attractive to generate such elevator inputs. Given the nonlinear aircraft’s linear time-invariant model based controllers at the inner loop, constant thrust aircraft navigation requires these controllers derived for sensor fusions. That is, in state feedback format, usually an identity matrix is considered for all state variables as measurements whereas in the present paper, particular combinations of the sensors in the feedback path are considered. A procedure to derive the stabilizing controllers that accommodates the sensor fusions is presented in partial pole placement framework. Then the nonlinear aircraft embedded with such controllers is simulated with capabilities to avoid the time critical obstacles. Examples by using a micro air vehicle model are illustrated.

Proportional–integral–derivative controller family for pole placement

In this paper, linear time-invariant square systems are considered. A procedure to design infinitely many proportional–integral–derivative controllers, all of them assigning closed-loop poles (or closed-loop eigenvalues), at desired locations fixed in the open left half plane of the complex plane is presented. The formulation accommodates partial pole placement features. The state-space realization of the linear system incorporated with a proportional–integral–derivative controller boils down to the generalized eigenvalue problem. The generalized eigenvalue-eigenvector constraint is transformed into a system of underdetermined linear homogenous set of equations whose unknowns include proportional–integral–derivative parameters. Hence, the proportional–integral–derivative solution sets are infinitely many for the chosen closed-loop eigenvalues in the eigenvalue-eigenvector constraint. The solution set is also useful to reduce the tracking errors and improve the performance. Three examples are illustrated.