Comparison of methods of nonlinear control systems design

The issue of designing nonlinear control systems is a complex problem. A lot of methods are known that allow us to find a suitable control for a given nonlinear object that provides asymptotic stability of the nonlinear system equilibrium and an acceptable quality of the transient process. Many of these methods are difficult to apply in practice. Thus, comparing some of the methods in terms of simplicity of use is of great interest. Two analytical methods for the synthesis of nonlinear control systems are considered. They are the algebraic polynomial-matrix method that uses a quasilinear model, and the feedback linearization method that uses the Brunovsky model in combination with special feedbacks. A comparative analysis of the algebraic polynomial-matrix method and the feedback linearization method is carried out. It is found out that the algebraic polynomial-matrix method (APM) is much simpler than the feedback linearization method (FLM). A numerical example of designing a system that is synthesized by these methods is considered. It is found out that the system synthesized by the APM method has a region of attraction of the equilibrium position twice as large as the region of attraction of the system synthesized by the FLM method. It is reasonable to use the algebraic polynomial-matrix method with the quasilinear models in case of synthesis of control systems of objects with differentiable nonlinearities.

Control device synthesis by polynomial matrix fractional descriptions method with limitation on controller structure

Modification of the algorithm for the polynomial synthesis of a multi-channel controller was proposed to preserve all control channels in this article. In order to test the functionality of the proposed modification, an example of a linear model of an unstable multi-channel plant is considered. The choice of the plant was determined by the possibility of a visual algorithm demonstration for polynomial synthesis of the controller, taking into account the proposed modifications.The plant was represented as three series-connected standard links: an aperiodic link of the first order, an unstable link, and an integrator, and has three input and two output channels. The control in the system is carried out in the feedback of the system and is summed up with the input impact. The feature of the plant is to limit the task to the second output, since it is essentially a derivative of the first output. In addition, the plant has a direct input–output channel. That is, the traversal matrix of the system is nonzero (when described through the state space). The synthesis task was set as follows: it is necessary to achieve certain quality indicators of the output vector value while maintaining all three control channels of the plant.

Polynomial matrix synthesis method of a subordinate control system

The main difference between controlsyn thesis approaches is the various mathematical representation of a plant or system model. The aim of the work is to represent a single channel control plant model by a multichannel one and to obtain an identical design result for a single channel multiloop synthesis method by a multichannel one. For these purposes, direct current motor model is used as an example of a single channel plant. Classical approach to design control system for that kind of plant is to describe it as a serial connected transfer functions and design a multiloop system in accordance with subordinate concept. Polynomial matrix synthesis method with Sylvester matrix is utilized to make identical subordinate regulator. By several transformations, polynomial matrix description was obtained, that describe the plant as one input and three output model and subordinate regulator as a three input and one output model. Arbitrary parameters of regulator were introduced for extended null placement.

Sum-of-squares chordal decomposition of polynomial matrix inequalities

We prove decomposition theorems for sparse positive (semi)definite polynomial matrices that can be viewed as sparsity-exploiting versions of the Hilbert–Artin, Reznick, Putinar, and Putinar–Vasilescu Positivstellensätze. First, we establish that a polynomial matrix P(x) with chordal sparsity is positive semidefinite for all $$x\in \mathbb {R}^n$$ x ∈ R n if and only if there exists a sum-of-squares (SOS) polynomial $$\sigma (x)$$ σ ( x ) such that $$\sigma P$$ σ P is a sum of sparse SOS matrices. Second, we show that setting $$\sigma (x)=(x_1^2 + \cdots + x_n^2)^\nu $$ σ ( x ) = ( x 1 2 + ⋯ + x n 2 ) ν for some integer $$\nu $$ ν suffices if P is homogeneous and positive definite globally. Third, we prove that if P is positive definite on a compact semialgebraic set $$\mathcal {K}=\{x:g_1(x)\ge 0,\ldots ,g_m(x)\ge 0\}$$ K = { x : g 1 ( x ) ≥ 0 , … , g m ( x ) ≥ 0 } satisfying the Archimedean condition, then $$P(x) = S_0(x) + g_1(x)S_1(x) + \cdots + g_m(x)S_m(x)$$ P ( x ) = S 0 ( x ) + g 1 ( x ) S 1 ( x ) + ⋯ + g m ( x ) S m ( x ) for matrices $$S_i(x)$$ S i ( x ) that are sums of sparse SOS matrices. Finally, if $$\mathcal {K}$$ K is not compact or does not satisfy the Archimedean condition, we obtain a similar decomposition for $$(x_1^2 + \cdots + x_n^2)^\nu P(x)$$ ( x 1 2 + ⋯ + x n 2 ) ν P ( x ) with some integer $$\nu \ge 0$$ ν ≥ 0 when P and $$g_1,\ldots ,g_m$$ g 1 , … , g m are homogeneous of even degree. Using these results, we find sparse SOS representation theorems for polynomials that are quadratic and correlatively sparse in a subset of variables, and we construct new convergent hierarchies of sparsity-exploiting SOS reformulations for convex optimization problems with large and sparse polynomial matrix inequalities. Numerical examples demonstrate that these hierarchies can have a significantly lower computational complexity than traditional ones.

Polynomial method for the synthesis of regulators for the special case of multichannel objects with one input variable and several output values

Currently, an urgent task in control theory is the synthesis of regulators for objects with a smaller number of input values compared to output ones, such objects are described by matrix transfer functions of a non-square shape. A particular case of a multichannel object with one input variable and two / three / four output variables is considered; the matrix transfer function of such an object has not a square shape, but one column and two / three / four rows. To calculate the controllers, a polynomial synthesis technique is used, which consists in using a polynomial matrix description of a closed-loop control system. A feature of this approach is the ability to write the characteristic matrix of a closed multichannel system through the polynomial matrices of the object and the controller in the form of a matrix Diophantine equation. By solving the Diophantine equation, the desired poles of the matrix characteristic polynomial of the closed system are set. There are many options for solving the Diophantine equation and one of them is to represent the polynomial matrix Diophantine equation as a system of linear algebraic equations in matrix form, where the matrix of the system is the Sylvester matrix. The choice of the order of the polynomial matrix controller and the order of the characteristic matrix is carried out on the basis of the theorem given in the works of Chi-Tsong Chen, which always holds for controlled objects. If the minimum order of the controller is chosen in accordance with this theorem, and the Sylvester matrix has not full rank, then this means that there are more unknown elements in the system of linear algebraic equations than there are equations. In this case, the solution corresponding to the selected basic minor has free parameters, which are the parameters of the regulators. Free parameters of regulators can be set arbitrarily, which is used to set or exclude some zeros in a closed system. Thus, using various examples of objects with a non-square matrix transfer function, a polynomial synthesis technique is illustrated, which allows not only specifying the poles of a closed system, but also some zeros, which is a significant advantage, especially when synthesizing controllers for multichannel objects.

Solving systems of matrix equations of the second degree

Matrix equations and systems of matrix equations are widely used in control system optimization problems. However, the methods for their solving are developed only for the most popular matrix equations – Riccati and Lyapunov equations, and there is no universal approach for solving problems of this class. This paper summarizes the previously considered method of solving systems of algebraic equations over a field of real numbers [1] and proposes a scheme for systems of polynomial matrix equations of the second degree with many unknowns. A recurrent formula for fractionalization a solution into a continued matrix fraction is also given. The convergence of the proposed method is investigated. The results of numerical experiments that confirm the validity of theoretical calculations and the effectiveness of the proposed scheme are presented.