Synthesis of Robust Linear Stationary Dynamic Systems Based on Improvement of the Controllability and Observability of the Computational Model of the Object

Abstract

An iterative method is developed for the synthesis of control systems of low parametric sensitivity with dynamic (polynomial) controllers, at each step of which singular numbers of controllability and observability gramians are intentionally changed, and a precontroller is formed that transforms the structure of the computational model of an object to increase its controllability and observability, taking into account the available opportunities and limitations. The method of polynomial modal control calculates the parameters of the main controller and evaluates the robust properties of the synthesized system in the given intervals of variation of the internal parameters.

APPENDIX

We synthesize a robust system for controlling the tension of an elastic long-length material in the zone of transportation of a production line using a PC.

The control object’s matrices, similar to those used earlier in [11], have the following form:

$${\mathbf{A}} \pm {{\Delta }}{\mathbf{A}} = \left[ {\begin{array}{*{20}{c}} { - 2~} \\ {0.667} \\ { - 7.5} \\ {~750 \pm 150} \end{array}\begin{array}{*{20}{c}} {~~~0} \\ {~~ - 0.667} \\ {~~ - 7.5} \\ {~~~750 \pm 150} \end{array}\begin{array}{*{20}{c}} {~~~2~} \\ {~~ - 0.667} \\ {~ - 8} \\ {~~~~0} \end{array}\begin{array}{*{20}{c}} {~~~ - 0.1 \pm 0.02~} \\ {~~0.033 \pm 0.007} \\ {~0} \\ { - 10} \end{array}} \right],$$

$$~{\mathbf{B}} = \left[ {\begin{array}{*{20}{c}} 0 \\ {\begin{array}{*{20}{c}} {0.667} \\ 0 \end{array}} \\ 0 \end{array}} \right],\quad {\mathbf{C}} = \left[ {1500~~~1500~~~0~~~0} \right].$$

We take the Newton polynomial D(s) = (s + 17.3)⁴ as the desired XP, which determines the requirements for the performance and quality of the control of the synthesized ACS at the nominal parameters of the CO. It is required to ensure the preservation of the aperiodic nature of transients with a permissible speed deviation within 10% for the given variations in the object parameters and 20% changes in the controller’s own parameters.

We assume that, in addition to the output signal, all coordinates of the state are available for measurement and additional effects on the derivatives of the first, second, and third coordinates are possible. Thus, binary matrix masks can be given in the form

$${{{\mathbf{E}}}_{1}} = \left[ {\begin{array}{*{20}{c}} 1 \\ 1 \\ 0 \\ 0 \end{array}\begin{array}{*{20}{c}} {~1} \\ {~1} \\ {~0} \\ {~0} \end{array}\begin{array}{*{20}{c}} {~1} \\ {~1} \\ {~1} \\ {~0} \end{array}\begin{array}{*{20}{c}} {~1} \\ {~1} \\ {~0} \\ {~0} \end{array}} \right],\quad {{{\mathbf{E}}}_{2}} = \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \\ 0 \\ 0 \end{array}} \right],\quad {{{\mathbf{E}}}_{3}} = \left[ {0~~0~~0~~0} \right].$$

The calculation of the TF of the control object gives the following expressions for its polynomials:

$$B\left( s \right) = {{10}^{3}}{{s}^{3}} + 2 \times {{10}^{4}}{{s}^{2}} + 1.46 \times {{10}^{5}}s + 4.6 \times {{10}^{5}},$$

$$A\left( s \right) = {{s}^{4}} + 20.667{{s}^{3}} + 249.33{{s}^{2}} + 1.397 \times {{10}^{3}}s + 2.91 \times {{10}^{3}}$$

with the corresponding distribution of zeros and poles shown in Fig. 5a.

The values of the indicators of controllability, observability, and degeneracy of the CO will be as follows:

$${\text{||}}{{{\mathbf{P}}}_{U}}{\text{|}}{{{\text{|}}}_{1}} = 1.5,~\quad ~{\text{||}}{{{\mathbf{P}}}_{V}}{\text{|}}{{{\text{|}}}_{1}} = 245.5,\quad {\text{||}}{{{\mathbf{P}}}_{{\bar {U}\tilde {U}}}}{\text{|}}{{{\text{|}}}_{1}} = 1958.9.$$

As a result of the calculation and singular value decomposition of the controllability and observability gramians, we obtain

$${{{\mathbf{\hat {\Sigma }}}}_{c}} = {\text{diag}}\left\{ {134.21,\,\,4.92 \times {{{10}}^{{ - 2}}},\,\,1.24 \times {{{10}}^{{ - 2}}},\,\,8.64 \times {{{10}}^{{ - 4}}}} \right\},$$

$${{{\mathbf{\hat {\Sigma }}}}_{o}} = {\text{diag}}\left\{ {3.88 \times {{{10}}^{5}},\,\,1.29 \times {{{10}}^{4}},\,\,8.64 \times {{{10}}^{2}},\,\,~1.56 \times {{{10}}^{{ - 1}}}} \right\}.$$

The obtained results give grounds to predict a low parametric roughness of the synthesized system.

Indeed, the calculation of the polynomials of the PC indicates its nonminimum-phase (unstable) character:

$$R\left( s \right) = 1.77{{s}^{3}} + 30.68{{s}^{2}} + 338.97s + 1577.04,$$

$$C\left( s \right) = {{s}^{3}} - 1.67 \times {{10}^{3}}{{s}^{2}} - 2.55 \times {{10}^{4}}s - 8.69 \times {{10}^{4}},$$

which, as shown by the corresponding transient characteristics in Fig. 6a, does not provide the desired robust properties of the synthesized ACS.

Fig. 6.

Transient response for systems 1–3, respectively, at the nominal parameters of the CO and at their maximum deviations in the positive and negative directions: (a) the original ACS with PC; (b) synthesized ACS; (c) ACS when varying the controller parameters within ±20%; (d) ACS after switching off the precontroller; (e) ACS after turning off the precontroller when varying the parameters of the controller within ±20%.

The iterative correction of the singular values of both gramians, performed by the developed method, makes it possible to significantly improve the system properties of the object,

$${{{\mathbf{\hat {\Sigma }}}}_{c}} = {\text{diag}}\left\{ {109.49,4.4 \times {{{10}}^{{ - 2}}},9.51 \times {{{10}}^{{ - 3}}},4.29 \times {{{10}}^{{ - 3}}}} \right\},$$

$${{{\mathbf{\hat {\Sigma }}}}_{o}} = {\text{diag}}\left\{ {2.54 \times {{{10}}^{5}},2.81 \times {{{10}}^{4}},2.65 \times {{{10}}^{3}},0.903} \right\},$$

$${\text{||}}{{{\mathbf{P}}}_{U}}{\text{|}}{{{\text{|}}}_{1}} = 1.5,\quad {\text{||}}{{{\mathbf{P}}}_{V}}{\text{|}}{{{\text{|}}}_{1}} = 60.98,\quad {\text{||}}{{{\mathbf{P}}}_{{\bar {U}\tilde {U}}}}{\text{|}}{{{\text{|}}}_{1}} = 1889.57,$$

by using the appropriate precontroller:

$${\mathbf{\hat {A}}} = \left[ {\begin{array}{*{20}{c}} { - 3.332}&{6.638}&{4.937}&{ - 0.339} \\ { - 2.204}&{ - 4.69}&{ - 1.258}&{0.037} \\ { - 7.478}&{7.502}&{ - 4.89}&{0.001} \\ {748.188}&{750.111}&{ - 0.198}&{ - 10.001} \end{array}} \right],$$

$${\mathbf{A}}' = \left[ {\begin{array}{*{20}{c}} { - 1.332}&{6.638}&{2.937}&{ - 0.239} \\ { - 2.871}&{ - 4.023}&{ - 0.591}&{0.004} \\ {0.022}&{15.002}&{3.11}&{0.001} \\ { - 1.812}&{0.111}&{ - 0.198}&{ - 0.001} \end{array}} \right],$$

$${\mathbf{A}}{\kern 1pt} '\, \circ {{{\mathbf{E}}}_{1}} = \left[ {\begin{array}{*{20}{c}} { - 1.332} \\ { - 2.87} \\ 0 \\ 0 \end{array}\begin{array}{*{20}{c}} {~~~~6.64} \\ {~ - 4.022} \\ {~0} \\ {~0} \end{array}\begin{array}{*{20}{c}} {~~~~2.937} \\ {~ - 0.591} \\ {~~~~3.11} \\ {~0} \end{array}\begin{array}{*{20}{c}} {~ - 0.239} \\ {~~~~0.07} \\ {~0} \\ {~0} \end{array}} \right].$$

This allows us to change the design model of the CO:

$$\hat {A}\left( s \right) = {{s}^{4}} + 22.91{{s}^{3}} + 471.95{{s}^{2}} + 3.01 \times {{10}^{3}}s + 8.25 \times {{10}^{3}},$$

$$\hat {B}\left( s \right) = 1000{{s}^{3}} + 2.49 \times {{10}^{4}}{{s}^{2}} + 2.71 \times {{10}^{5}}s + 1.23 \times {{10}^{6}},$$

as well as the distribution of zeros and poles (Fig. 5b), and form a new control device with TF polynomials:

$$\hat {R}\left( s \right) = 9.5 \times {{10}^{{ - 2}}}{{s}^{3}} + 3.07{{s}^{2}} + 63.66s + 352.24,$$

$$\hat {C}\left( s \right) = {{s}^{3}} + 3.53{{s}^{2}} + 304.94s + 3.78 \times {{10}^{3}},$$

providing a complete solution of the problem in relation to the distribution of poles (Fig. 5c) and achieving the specified robust properties of the synthesized ACS (Figs. 6b, 6c).

Taking into account the complexity of measuring the coordinates of the CO state and focusing on the capabilities of the SCMO method, we exclude the formed precontroller from the composition of the synthesized ACS. The new distribution of zeros and poles of the system, which is formed after the precontroller is turned off, is shown in Fig. 5d.

From the analysis of the corresponding transient characteristics of the ACS (Figs. 6d, 6e) obtained with the given variations in the parameters of the object and the controller itself, synthesized according to the improved CO model, it follows that, despite the reduction of the control device, the system largely retains the achieved quality indicators and provides a solution to the problem of robust control.