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This paper considers controller parameterization method of

A lot of control problems can be modeled as or transformed approximately to polynomial control systems, which are of great significance in the nonlinear theory. In recent years, a series of achievements have been obtained via symbolic computation and engineering applications [

As we all know, the internal stability and the external disturbance attenuation are the basic constraints of control systems. However, it is critical to satisfy some desired control objectives in designing a practical control system. It is a sophisticated and efficient way to find the parameterized controller to solve complex control problems, namely, the

The controllers obtained in [

In order to avoid solving HJI inequalities (or equations) and obtain a parameterized controller, this paper presents a novel, straightforward, and convenient method to parameterizing controller for PHSs and gives an algorithm for solving parameters of the controller by symbolic computation. The main results of the paper are as follows.

A controller with parameters is obtained via the approach to parameterizing controller for PHSs. Because the Hamiltonian function can be used to build a Lyapunov function of a dissipative Hamiltonian system, the approach obtained here avoids solving HJI inequalities (or equations). Moreover, the obtained controller with parameters has a simpler form than the controller obtained in [

Parameters ranges are obtained via an algorithm based on symbolic computation. The controller with parameters is effective to

The remainder of this paper is organized as follows. In Section

Consider the following polynomial Hamiltonian system with dissipation [

Note that

The problem considered in this paper is to propose an approach to parameterizing controller for systems (

In the end, we give a definition and a lemma required in the next section.

System (

Consider a nonlinear system

In this section, we propose an

Suppose Assumption

Consider the candidate Lyapunov function

Hence, the closed-loop system converges to the largest invariant set, which is contained in

From Assumption

On the other hand, since

(1) As compared with the controller proposed in [

(2) Parameters ranges of polynomial vector

(3) Wang et al. [

From condition (

S1. Set

S2. Let

S3. The influence of high order items can be ignored because this paper considers locally asymptotically stable for system. Choose all terms of

S3.1. Observe equations

S3.2. Obtain a set of parameters solution

S3.3. Substitute

S4. Rewrite

S4.1. Observe inequalities

S4.2. Obtain a set of parameters solution

S5. Let

(1) The algorithm starts from

(2) The CAD algorithm is given by Semi-Algebraic-Set-Tools of Regular-Chains in Maple 16.

(3) It is merely to simplify computation that we let some parameters be zero before using CAD algorithm. However, these parameters are not necessarily zero. So the set of parameters solution obtained by the algorithm is a subset of solutions.

Consider a polynomial Hamiltonian system with dissipation (

From system (

Then, we check that condition (

Next, we consider condition (

We know

From system (

Let

All principal minors of

In order to evaluate the robustness of the controller (

Suppose that

The simulation results are shown in Figures

Swing curves of

Swing curves of

Swing curves of

Swing curves of

In order to evaluate the robustness optimization of the system by adjusting the parameters of controller (

Suppose that

Swing curves of

Swing curves of

From Figure

In this paper, an approach to parameterizing controller for polynomial Hamiltonian systems has been considered. A controller with parameters has been obtained using Hamiltonian function method and an algorithm for solving parameters of the controller has been proposed with symbolic computation. The proposed parameterization method avoids solving Hamilton-Jacobi-Isaacs equations and thus the obtained controllers with parameters are easier as compared to some existing ones. The numerical experiment and simulations show that the controller is efficient in

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China (nos. 61074189 and 61374001).

_{∞}output feedback controllers

_{∞}controllers with full information for dissipative Hamiltonian systems