ROBUSTNESS ANALYSIS OF SOME MULTIVARIABLE FEEDBACK CONTROL SYSTEMS CONTAINING GENERALIZED INVERSE MODEL-BASED CONTROLLERS



AZARSKOV V.N., Zhiteckii L.S., SOLOVCHUK K.Yu. The work deals with utilizing the so-called generalized inverse model concept advanced before by the authors to design the robust linear discrete-time feedback controllers for controlling both nonlinearand linear multivariable static plants. The robustness conditions of the closed-loop control systems are established.

Робота стосується використання так званої концепції узагальненої оберненої моделі, висунутої раніше авторами, для побудови робастних лінійних дискретних регуляторів в ланцюгу зворотного зв’язку для керування як нелінійними, так і лінійними багатовимірними статичними об’єктами. Встановлено умови робастності замкнутих систем керування.

UDC 681.5

Introduction

The design of an advanced controller for MIMO (multi-input multi-output) processes requires a model giving some description of the process behavior. Within the framework of this concept, the inverse model-based controller was reported in [1] by one of the authors together with his colleagues to dealing with a linear discrete-time multivariable process control. To implement the method of [1], the gain matrix of a plant model needs to be non-singular. However, this matrix may actually be singular, in general.

Recently, the so-called generalized inverse (pseudo-inverse) model-based control concept was advanced in [2] to cope with the singularity which may appear in the gain matrix. This concept makes it possible to implement the perfect control of linear MIMO plants with nouncertainties in their description. It turns out that the generalized inverse model-based controller can be robust even if the plant is unknown and nonlinear[3].

This work is an extension and generalization of robustnes sresults established in [3].

Problem Statement

Consider the discrete-time multivariable static plant described by the equation

${{y}_{n}}=\phi ({{u}_{n-1}})+{{d}_{n}}$ (1)

where $y={{[{{y}^{(1)}},\ldots,\ {{y}^{(m)}}]}^{T}}$ denotes the m-dimensional output vector, $u={{[{{u}^{(1)}},\ldots ,\ {{u}^{(r)}}]}^{T}}$ denotes the r-dimensional input (control) vector, ${{d}_{n}}={{[d_{n}^{(1)},\ldots,\ d_{n}^{(m)}]}^{T}}$ is unmeasurable disturbance vector, and $\phi :\ \ {{\mathbf{R}}^{r}}\to{{\mathbf{R}}^{m}}$ represents some nonlinear (in general) vector-valued function

\[\phi(u)={{[\phi _{{}}^{(1)}(u),\ldots ,\ \phi _{{}}^{(m)}(u)]}^{T}}.\quad\quad\quad(2)\]

Suppose that the number of inputs is not more than the number of outputs, i.e., $r\le m.$

Denoting ${{b}^{(ij)}}(u):=\partial{{\phi }^{(i)}}(u)/\partial {{u}^{(j)}},$ introduce the matrix

Let ${{y}^{*}}:={{[{{y}^{*(1)}},\ldots,\ {{y}^{*(m)}}]}^{T}}$ $({{y}^{*(i)}}\equiv \text{const})$ be some nonzero vector defining the desired output vector (a given set-point).

As in [3], we choose a fixed $m\times r$ matrix ${{B}_{0}}=(b_{0}^{(ij)})$ whose elements satisfy

\[b_{\min }^{(ij)}\le b_{^{0}}^{(ij)}\le b_{\max }^{(ij)}\]

and determine the so-called generalized inverse matrix $B_{0}^{+}=(\beta _{0}^{(ij)})$ as

\[B_{0}^{+}=\underset{\delta\to 0}{\mathop{\lim }}\,\ B_{0}^{T}{{(B_{0}^{T}B_{0}^{{}}+{{\delta}^{2}}{{I}_{r}})}^{-1}},\]

where ${{I}_{r}}$ denotes the $r\times r$ unit matrix.

Following to [3], the generalized inverse model-based control law will be designed in the form

\[{{u}_{n}}={{u}_{n-1}}+B_{0}^{+}({{y}^{*}}-{{y}_{n}}).\quad\quad\quad(6)\]

The problem is to give the conditions guaranteeing the robustness properties of the feedback control system (1), (6) for any $\phi (u)$ which yields $B(u)$ whose elements satisfythe restriction (4).

Main Results

Before going tothe robustness analysis of the control system (1), (6) we note that the equilibrium state, $({{u}^{\text{e}}},\ {{y}^{\text{e}}}),$ of this systemis defined as a solution $u={{u}^{\text{e}}}$ of the equation

\[B_{0}^{+}({{y}^{*}}-\phi (u))={{0}_{r}}\quad\quad\quad (7)\]

in which ${{0}_{r}}$ denotes $r$-dimensional zero vector.

Introduce the variables ${{\sigma }^{(ik)}}$ defined by

\[{{\sigma}^{(ik)}}=\sum\limits_{j=1}^{m}{{{\beta }^{(ij)}}}{{\delta }^{(jk)}}\]

with ${{\delta }^{(ij)}}$ satisfying the restriction

\[b_{\min }^{(ij)}-b_{0}^{(ij)}\ \le\ {{\delta }^{(ij)}}\ \le \ b_{\max }^{(ij)}-b_{0}^{(ij)},\]

\[i=1,\ldots ,\ m,\ j=1,\ldots ,{r}. \quad\quad\quad(8)\]

Let the equilibrium state given by (7) exist. Then, the robustness analysis problem canbe shown to be formulated as the following linear programming problems:

find

\[\min {{\sigma }^{(ik)}}, \max {{\sigma }^{(ik)}}\quad\quad\quad(9)\]

under the restrictions (8).

Consider the nonlinear case assuming that the matrix ${{B}_{0}}$ to be chosen by the designer is a matrix of the full rank, i.e., $\text{rank }{{B}_{0}}=r.$ It turns out that in this case, the conditions

\[\sum\limits_{k=1}^{r}{\max \{|\min {{\sigma}^{(ik)}}|,}\ |\max {{\sigma }^{(ik)}}|\}\quad\quad\quad (10)\]

in which $\min {{\sigma }^{(ik)}}$ and $\max {{\sigma }^{(ik)}}$ represent the solutions of the linear programming problems (8), (9), are the sufficient conditions for the robustness of the controller (6).

Conclusions

In this work we show that the robustness properties of the discrete-time closed-loop control system which contains a multivariable static (linear or nonlinear) plant and the linear generalized inverse model-based controller will be guaranteed if certain conditions on its model are satisfied. These conditions can easy be verified.

References

$1.$ Skurikhin V.I., Protsenko N.M., Zhitetsky L.S. Multiple-connected systemof technological processes control with table of objects // Proc. IFAC 3rd Multivariable Technological Systems Symposium (Manchester, UK), 1974.– P.S35-1–S35-4.

$2.$ Azarskov V.N., Zhiteckii L.S., Solovchuk K.Yu. Discrete-time control oflinear multivariable systems with either singular or ill-conditioned transfer function matrices // Proceedings of the National Aviation University, 2014.– No2.– P. 19-27.

$3.$ Zhiteckii L.S., Azarskov V.N., Solovchuk K.Yu., Sushchenko O.A. Discrete-time robust steady-state control of nonlinear multivariable systems: aunified approach // Preprints 19th IFAC World Congress. Cape Town, South Africa, 2014.– P. 8140-8145.

Jun 21, 2016