Pole Constraints Research Articles

Robust [formula omitted] filter design with pole constraints for discrete-time systems

The problem of full-order guaranteed cost H ∞ filtering design for discrete-time systems with polytope type uncertainty is investigated in this paper. The main purpose is to obtain a stable and proper linear filter such that the filtering error system remains robustly stable within a prespecified H ∞ attenuation level. Sufficient conditions for the existence of such robust filter are provided in terms of linear matrix inequalities, which can be efficiently solved by means of high-performance convex optimization procedures with global convergence assured. Moreover, as an improvement, the filter dynamics can be constrained to some specific regions inside the unit open disk.

Robust control of linear uncertain systems with regional pole and variance constraints

This paper concerns the design problem of state feedback controllers which guarantee the closed-loop poles within a specified disc and steady-state variances to be less than a set of given upper bounds for linear systems with norm-bounded parameter uncertainties. Using the linear matrix inequality approach, the existence conditions of such controllers are derived. A parametrized representation of the desired controllers is presented in terms of the feasible solutions to a certain linear matrix inequality system. Based on this, a solution to the minimum-effect guaranteed-performance design problem is presented in the sense that the required control effort is minimized subject to performance constraints.

Robust H∞ control for uncertain discrete-time systems with circular pole constraints

This paper investigates the problem of robust H ∞ control for uncertain discrete-time systems with circular pole constraints. The system under consideration is subject to norm-bounded time-invariant uncertainties in both the state and input matrices. The problem we address is to design state feedback controllers such that the closed poles are located within a prespecified circular region, and the H ∞ norm of the closed-loop transfer function is strictly less than a given positive scalar for all admissible uncertainties. By introducing the notion of quadratic d stabilizability with an H ∞ norm-bound, the problem is solved. Necessary and sufficient conditions for quadratic d stabilizability with an H ∞ norm-bound are derived. Our results can be regarded as extensions of existing results on robust H ∞ control and robust pole assignment of uncertain systems.

Robust state estimation for perturbed systems with error variance and circular pole constraints: The discrete-time case

This paper is concerned with the problem of robust state estimation for linear perturbed discrete-time systems with error variance and circular pole constraints. The goal of this problem addressed is the design of a linear state estimator such that, for all admissible uncertainties in both state and output equations, the following two performance requirements are simultaneously satisfied: (1) the poles of the filtering matrix are all constrained to lie inside a prespecified circular region; and (2) the steady-state variance of the estimation error for each state is not more than the individual prespecified value. It is shown that this problem can be converted to an auxiliary matrix assignment problem and solved by using an algebraic matrix equation/inequality approach. Specifically, the conditions for the existence of desired estimators are obtained and the explicit expression of these estimators is also derived. The main results are then extended to the case when an H performance requirement is added. Finally, a numerical example is presented to demonstrate the significance of the proposed technique.

Diving autopilot design for underwater vehicles using multi-objective control synthesis

This paper presents the mathematical modeling, guidance and robust control synthesis of a highly maneuverable submersible vehicle (or underwater vehicle) when performing a specific mission at shallow submergence conditions. First, the vertical plane motions (heave and pitch) of the vehicle are modeled by a set of maneuvering equations. After model simplification, a state-space model is compactly obtained. Then a state-feedback controller is proposed for the accurate depth-keeping and pitch motion controls of the vehicle. The control actions to the generalized plant can be provided by the mixedH2/H∞ optimal synthesis as well as closed-loop pole constraint with LMIs. The feasibility of the guidance and control approach is verified with direct numerical simulations. The proposed approach ensures reasonable depth-keeping and minimal pitch motions, even under a given uncertainty condition.

Vehicle dynamics and control synthesis for four-wheel steering passenger cars

This paper is concerned with the active robust autopilot design of a four-wheel steering vehicle against external disturbances. Firstly, the effect of four-wheel steering and independent wheel torques for lateral/directional and roll motions is modelled by a set of linear models under proper manoeuvring conditions. To enhance the dynamic performance of an automobile system, a mixed H2/H∞ synthesis with pole constraint is designed on the basis of full state feedback applying linear matrix inequality (LMI) theory. For lateral/directional and roll motions, the steering angles are actively controlled by steering wheel angles through the actuator dynamics. The wheel power and braking are also controlled by independent wheel torques. Simulation results indicate that the proposed control approach can achieve predetermined performance (or acceptable level of disturbance attenuation) and stability as well as robustness even when external disturbances are severe. The active 4WS car along with steering and wheel torque control algorithms allows greater manoeuvrability and improved stability in a wide range of uncertainty.

Robust H∞ State Feedback Control With Regional Pole Constraints: An Algebraic Riccati Equation Approach

This paper focuses on the controller design for uncertain linear continuous-time systems with H∞ norm and circular pole constraints and addresses the following multiobjective simultaneous realization problem: designing a state feedback controller such that the closed-loop system, for all admissible parameter uncertainties, simultaneously satisfies the prespecified H∞ norm constraint on the transfer function from disturbance input to output and the prespecified circular pole constraint on the closed-loop matrix. An effective, algebraic, modified Riccati equation approach is developed to solve this problem. The existence conditions, as well as the analytical expression of desired controllers, are derived. A numerical example is provided to show the directness and effectiveness of the present approach.

Robust state estimation for perturbed continuous systems with circular pole and error variance constraints

In this paper, the problem of circular pole and variance-constrained state estimator design is studied for perturbed linear continuous-time systems. The system under consideration is subject to unstructured time-invariant norm-bounded parameter perturbations in both the state and measurement matrices. The problem we address is the design of a state estimator such that, for all admissible time-invariant perturbations, the poles of the steady-state filtering matrix are assigned inside a prespecified circular region, and the variance of the estimation error of each state is not more that the individual prespecified value, simultaneously. An algebraic Riccati-like matrix inequality approach is proposed to solve the above problem. Specifically, the existence conditions and an explicit expression for the desired estimators are obtained. Furthermore, an illustrative example is presented to demonstrate the effectiveness of the proposed design procedure.

Robust controller design for uncertain linear systems with circular pole constraints

In this paper, the problem of robust controller design for uncertain linear continuous-and discrete-time systems with circular pole constraints is considered. The problem we address is the design of robust state feedback controllers such that all poles of the closed-loop systems are placed within a prescribed circular region. Sufficient conditions for the existence of the desired controllers are given based on the generalized inverse theory. The analytical expression of the set of desired controllers is also presented. It is emphasized that, when the uncertainties are absent, the results include the characterization of all state-feedback controllers achieving the specified circular pole constraints. An illustrative example is given to show the applicability of the proposed design procedure. Accepted in final revised form 7 February 1996. Communicated by Professor H. Kimura.

Linear quadratic optimal output feedback control for systems with poles in a specified region

The problem of output feedback linear quadratic regulator synthesis with regional pole constraints is studied. The synthesis algorithm is presented and its convergence is also proved. The effectiveness of the algorithm is illustrated by an example.

Optimal control with regional pole constraints via the mapping theory

The design of optimal linear, time-invariant systems with regional pole constraint is studied. The problem is to find the static state feedback controller to ensure that all closed-loop system poles lie inside the desired region H and, meanwhile, to minimise a multiobjective performance index. The desired region H can be represented by several inequalities. For some special cases, H may consist of several disjoint subregions. The performance index consists of two parts. One part is used to penalise the sustained error, and the other part is used to guarantee that the optimal solution will not occur on the boundary of the admissible controller set and to improve the robustness property of the closed-loop system. The necessary and sufficient condition for the existence of the admissible controller is found. The necessary condition that the optimal control law must be satisfied is derived. Furthermore, the robustness analysis of the regional pole restriction under unstructured perturbation is studied. Based on the Gersgorin's theorem a new method is presented, which calculates the allowable bounds of the unstructured perturbation, so that all the perturbed poles will still remain inside some regions.

Controller design for continuous systems with variance and circular pole constraints

The problem of controller design with variance and circular pole constraints is considered in this paper. The motivation for this problem is twofold: (1) many engineering systems have performance requirements naturally stated in terms of upper bounds on the variances of the system states; and (2) the fine transient properties of the closed-loop system are required in practice. The goal of this problem is to design the controller such that the closed-loop system meets the prespecified variance constraints and circular pole constraints simultaneously. It is shown that a desired controller is determined by using the solution of a modified Lyapunov equation. Necessary and sufficient conditions are given for the existence of desired controllers. The set of desired controllers (if they exist) is also characterized.

H2optimal control with an a-shifted pole constraint

Modified Lyapunov equations for regional pole constraints in H2 synthesis have been considered in recent papers. This approach involves a constraint equation for enforcing the pole constraints and leads to an auxiliary cost that overbounds the original H2 performance. The auxiliary cost can then be used for optimization with respect to the modified Lyapunov equations that characterize the constraint region. In this paper, we consider an α-shifted constraint region and show that an augmented system whose order is twice that of the original system can be constructed to eliminate the need for an overbound so that exact H2 cost optimization can be performed. This augmented system is then utilized for closed-loop controller synthesis within a decentralized static output feedback setting. The construction of the augmented system is a refinement of the approach of Gu et al. (1992). A numerical algorithm based upon the BFGS quasi-Newton method is used for computing the optimal controller gains. The numerical resu...

Controller design with regional pole constraints: Hyperbolic and horizontal strip regions

(24) where x2 is given at Eq. (23) and ( )' = d( )/d/x. This equation is valid to 0(e) as long as the flow stays in region 2a (Fig. 1), and it can be shown that if jii(0)>0 and e>0, then trajectories originating in region 2a remain in that region.9 Numerical comparisons show that solutions to Eq. (24) agree quite well with the exact solution to Eqs. (2) and (3). In this example there is only one region of phase space where the unperturbed solution is periodic. In case there is more than one such region [e.g., when V(x; ju) is quartic], then the form of the e = 0 solution and, hence, the form of the right-hand side of Eq. (14) are different in different regions. When the slow flow passes from one region to another, the averaged equation may lose validity. This is because the transition may involve crossing an instantaneous separatrix of the unperturbed system. At a separatrix, the period of the e = 0 solution becomes infinite, so that the average computed in Eq. (13) is over an infinite time interval, violating the conditions of the averaging theorem (see Ref. 9 for further discussion of separatrix crossing). Conclusions We have presented a general formulation for application of the method of averaging to a specific class of nonlinear equations. The method exploits the existence of an energy integral (the Hamiltonian) for the unperturbed system and leads to a single first-order equation for the slow evolution of the Hamiltonian. By using the canonical coordinate x as the fast variable, the need to identify the rapidly varying phase angle (as in Kruskal's method) is eliminated. As shown in the example, application is relatively straightforward when the form of the potential leads to an explicit solution to the unperturbed problem. References

Controller design with regional pole constraints

A design procedure is developed that combines linear-quadratic optimal control with regional pole placement. Specifically, a static and dynamic output-feedback control problem is addressed in which the poles of the closed-loop system are constrained to lie in specified regions of the complex plane. These regional pole constraints are embedded within the optimization process by replacing the covariance Lyapunov equation by a modified Lyapunov equation whose solution, in certain cases, leads to an upper bound on the quadratic cost functional. The results include necessary and sufficient conditions for characterizing static output-feedback controllers with bounded performance and regional pole constraints. Sufficient conditions are also presented for the fixed-order (i.e. full- and reduced-order) dynamic output-feedback problem with regional pole constraints. Circular, elliptical, vertical strip, parabolic, and section regions are considered. >

Optimal compensators with pole constraints

In an observer-based control system, the design freedom which remains after pole assignment is used to minimize-in an average sense-a given quadratic performance index on the plant state and control. The class of controllers and observers satisfying pole constraints is characterized through a certain parameter vector. An algorithm is then developed to compute the optimal controller-observer set. An example is worked out to illustrate that the Optimal controller based on estimated state is not, in general, optimal when the true state is used.