Linlin Ou

Low-Order Stabilization of LTI Systems With Time Delay

This paper considers the problem of stabilizing a single-input-single-output (SISO) linear time-invariant (LTI) plant with known time delay using a low-order controller, such as a Proportional (P), a Proportional-Integral (PI), or a proportional-integral-derivative (PID) controller. For the SISO LTI system with time delay, the closed-loop characteristic function is a quasipolynomial that possesses the following features: all its infinite roots are located on the left of certain vertical line of the complex plane, and the number of its unstable roots is finite. Necessary and sufficient conditions for the stability of LTI systems with time delay are first presented by employing an extended Hermite-Biehler Theorem applicable to quasi-polynomials. Based on the conditions, analytical algorithms are then proposed to compute the stabilizing sets of P, PI and PID controllers. The resulting characterizations of the stabilizing sets for P, PI and PID controllers are analogous to the Youla parameterization of all stabilizing controllers for plants without time delay. Numerical examples are provided to illustrate the proposed algorithm.

The local control scheme for switching consensus value in multi-agent systems

A local control strategy is presented to switch the consensus value for the first-order multi-agent system. When the local proportional controller is employed in the multi-agent systems, the Laplacian matrix of the system is changed. It is proved that the changed Laplacian has the same properties as the Laplacian matrix of the original multi-agent system for the consensus. Based on this, the parameter of the local controller, which can guarantee that all the agents change the original consensus value into the desired one, is determined in terms of the matrix calculation and the stability criterion. In practice, the control system must be implemented in the discrete form. Thus, the influence of the sampling period on the stability of the discrete multi-agent system with the local controller is analysed. The simulation results show the validity of the proposed method.

Robust adaptive tracking control of MIMO nonlinear systems in the presence of actuator hysteresis

Adaptive tracking control of a class of MIMO nonlinear system preceded by unknown hysteresis is investigated. Based on dynamic surface control, an adaptive robust control law is developed and compensators are designed to mitigate the influences of both the unknown bounded external uncertainties and the unknown Prandtl–Islinskii hysteresis. By adopting the low-pass filters, the explosion of complexity caused by tedious computation of the time derivatives of the virtual control laws is overcome. With the proposed control scheme, the closed-loop system is proved to be semi-globally ultimately bounded by the Lyapunov stability theory, and the output of the controlled system can track the desired trajectories with an arbitrarily small error. Finally, numerical simulations are given to verify the effectiveness of the proposed approach.

Stabilizing sets of the low-order controllers for the systems with multiple time delays

The low-order stabilization of the systems with multiple time delays is more difficult compared with that of the rational system and the system with single time delay. For an arbitrarily-given LTI (linear time-invariant) multiple time-delay plant, the stabilization problem of the Proportional (P) and Proportional-Integral (PI) is solved in this paper. Based on the generalized Nyquist criterion applicable to the quasipolynomial, analytical algorithms to determine the entire stabilizing sets of the P and PI controllers are presented, which avoids the delay approximation. The control gains chosen in the resulting stabilizing set can ensure the stability of the closed-loop system. The results demonstrate that, the characterizations of the stabilizing P and PI controllers are in the closed form and quasi-closed form, respectively. The proposed method is applicable to stable, unstable or non-minimum phase system with multiple time delays. Numerical example is provided to illustrate the validity of the proposed algorithms.

H ∞ robust design of PID controllers for arbitrary-order LTI systems with time delay

For a given arbitrary-order LTI (linear time-invariant) plant with time delay, we propose a directly parametric design method of the H∞ PID controller in an analytical manner. The design problem of the PID controllers satisfying the H∞ norm requirement is first cast into simultaneous stabilization problem of a family of complex quasipolynomials and the characteristic equation. Then, the linear programming characterization of the PID controllers that can ensure the stability of the complex quasipolynomials are developed on the basis of the extended Hermite-Biehler Theorem.Finally, The admissible set of the H∞ PID controllers is presented by combining such the results with the stabilizing set of the PID controllers. The results reveal that the set of the integral and derivative gains is a union of convex sets for a fixed proportional gain. The proposed scheme works without any approximation and enable one to find a set of the PID parameters satisfying the H∞ norm requirement conveniently.

On the pole of non-square transfer function matrix Moore–Penrose pseudo-inverses

An essential step in many controller design approaches is computing the inverse of the plant. For a square plant, its inverse is stable if the plant is minimum phase (MP). Nevertheless, this conclusion does not hold for a non-square plant. In this paper, the pole feature of the Moore–Penrose pseudo-inverse of a non-square transfer function matrix is analysed. Instead of complicated advanced mathematical tools, only basic results of polynomial theory and the Binet–Cauchy theorem are used in the analysing procedure. The condition for testing the stability of the Moore–Penrose pseudo-inverse of an MP non-square transfer function matrix is given. This condition implies that the Moore–Penrose pseudo-inverse of a non-square transfer function matrix cannot be directly used as the optimal controller. Numerical examples are provided to illustrate the correctness of the condition.

Disturbance observer-based control for consensus tracking of multi-agent systems with input delays from a frequency domain perspective

Abstract The consensus tracking controller design of multi-agent systems with diverse input delays is studied in this paper. A universal block diagram is established to describe the linear multi-agent systems based on transfer functions. The stabilization of the whole system is decomposed into the zero steady-state error control problem of each independent agent. A sufficient and necessary condition is accordingly deduced to impose on each controller. Based on the H 2 performance index of each subsystem, both the optimal consensus controller and disturbance observer (DOB) are derived analytically. The distributed H 2 DOB-based consensus controller can not only achieve consensus tracking for the reference input, but also reject the effects of external disturbance and model uncertainty. Some simulations are performed to illustrate the validity of the proposed design approach.

Stability region of fractional-order PI λ D μ controller for fractional-order systems with time delay

A simple and effective method to determine the region of fractional-order PI^λD^μ controllers that can stabilize a given fractional-order system with time delay is proposed in this paper. For each known proportional, integral or derivative gain in the PI^λD^μ controllers, the stability region with respect to the other two control gains is derived. Firstly, the boundaries of the fractional-order PI^λD^μ controllers are determined by using the D-decomposition method. Then, an analytical approach is presented to judge which region is the stability one among a lot of areas divided by the resultant boundaries. In comparison with other relevant methods, the main advantage of the proposed method lies in that it can effectively avoid choosing one point from each divided area and finding the stability region of the fractional-order PID controller by testing the system stability corresponding to each chosen point. Moreover, a special phenomenon is revealed: if λ + μ ≠ 2, the boundaries of the stability region in k_i-k_d plane are the curves for a given k value; otherwise, the stability regions in k_i-k_d plane are convex polygons. A numerical example is presented to check the validity of the proposed method. The proposed method can be applied to the fractional-order system free of the detailed model and only the frequency response data of the fractional-order system is required.

Optimal selection strategy for multi-agent system with single leader

In this paper, we consider a class of controlled consensus problem, where a subset of agents called informed agents can receive the information from a leader with constant state. First, we derive sufficient conditions for guaranteeing all the agents reach consensus on the leader state under fixed and switched topology, respectively. Second, we investigate the problem of selecting informed agents for yielding minimal time controlled consensus using optimization tool. Using YALMIP toolbox, we also find that the optimality of the agents could be characterized by some important index in complex network. Finally, we offer some numerical examples to illustrate the results.

Two-degree-of-freedom low-order control scheme for multi-agent systems

Two-degree-of-freedom distributed low-order control scheme which can both realize the global tasks and provide good performance for the multi-agent systems is proposed in this paper. The two-degree-of-freedom control protocol which contains the individual controllers and the coupling controllers is firstly presented. The individual and coupling controllers are both of the proportion-integration-differentiation (PID) type. The two-degree-of-freedom multi-agent system is established according to the proposed protocol. Such the system is decoupled to several single-input-single-output (SISO) subsystems based on matrix theory. For each SISO subsystems, the PID stable region of the individual controllers satisfying H∞ index is obtained based on the extended Hermite-Biehler theorem so as to improve the performance of each agent. Then, the control parameters of the coupling controllers that ensure the fast consensus of the whole system is determined by first deriving the stable region and then searching for the minimum of the real part of the roots of the characteristic equation in the obtained stable region. The simulation result verifies the effectiveness of the proposed method. The proposed two-degree-of-freedom low-order control scheme is applicable to the systems with arbitrary-order stable or unstable agents.