Parametric Identification for Robust Control

Abstract

Many contemporary automatic control applications require parametric identification, taking into account results of estimation errors unavoidable in practice. The subject of this chapter constitutes a procedure enabling effective calculation of optimal, in the sense of minimal expectation of losses, estimator of a parameter in the case when losses resulting from negative and positive errors have significantly different influence on system operation, which forces the use of an asymmetric loss function. The conditional version of the issue is also considered in detail, as the process is significantly dependent on factors available metrologically during the system operation. This material has been provided in ready-to-use form, not requiring additional studies and bibliographical exploration. The presented concept is directed towards robust control problems. As an appendix, the analysis of solutions of differential equations with discontinuous right-hand side, subordinated for this purpose, has been presented, which may be useful for describing the dynamics of such resultant systems.

Appendix (Solutions of Differential Equations with Discontinuous Right-Hand Side)

In the first example of Subchapter 2 the issue of the different types of solutions of differential equations, needed to describe the phenomena appearing in the robust control systems (e.g. sliding trajectories or over-regulations) depending on the relationships between the values of the parameter occurring in the object and the estimator assumed in the control algorithm, was mentioned. The classic solution of differential equations in many such cases does not exist or improperly describes the modeled reality, which frequently leads to theoretical errors and interpretative misunderstandings. In this appendix, the fundamental definitions and relations connected with the issue of solutions of differential equations with discontinuous right-hand side, which can be applied to the description of the dynamics of objects submitted to robust control, will be presented.

Consider the differential equation in the form commonly used in control engineering, i.e. with the control u distinguished:

$$ \dot{y}(t) = g\left( {y(t),u(t),t} \right), $$

(82)

where $y:T \to {\mathbb{R}}^{n}$, $u:T \to {\mathbb{R}}^{m}$, $g:{\mathbb{R}}^{n} \times {\mathbb{R}}^{m} \times T \to {\mathbb{R}}^{n}$, and $T \subset {\mathbb{R}}$ is a time interval with nonempty interior. Equation (82) is inherently connected with the initial condition

$$ y\left( {t_{0} } \right) = y_{0} , $$

(83)

while $t_{0} \in T$ and $y_{0} \in {\mathbb{R}}^{n}$ are fixed.

The classic solution of the differential equation is the function y, differentiable and fulfilling the Eq. (82) for every $t \in T$ (if the boundary of the interval T belongs to it, then a proper one-sided derivative is considered) as well as the condition (83). This solution is unique, if each classic solution is a function identically equal to it. The above definitions are fully consistent with intuition, at least within the scope of the basic problems and aspects concerning dynamic systems.

The solution thus defined can be useful when the functions u and g are continuous. The continuity of the right-hand side of the differential equation constitutes the sufficient condition for the existence of the classic solution, and although it is not a necessary condition, the assumption of continuity is practically inherent in considerations of solutions of this type. In the case of a finite number of discontinuities of the first type (i.e. when in the point of discontinuity there exist bounded one-sided derivatives, not necessarily equal to each other nor the value of the function at this point) of the functions u and g (in the case of the second, with respect to the independent variable t), the above definition can be supplemented by a somewhat informal treatment known as “joining” of solutions. Namely, the interval T is then divided into subintervals, in which the functions u and g are continuous, and in these subintervals classic solutions are found, after which these solutions are linked together (“joined”) with maintaining the continuity of the solution y (in the point of these discontinuities of the functions u and g with respect to variable t, the derivative $\dot{y}$ does not exist, due to the lack of equality of the one-sided derivatives). The above concept has a clear physical interpretation, one can then treat that at the moment of discontinuity of the functions u or g, the process described by the differential equation (82) begins again, with the initial condition (83) which is the left-hand limit of the solution from the previous area of continuity. From the mathematical point of view, such a procedure has a very limited scope of applicability to the case of the finite number of discontinuities of the functions u and g with respect to the independent variable.

A typical example illustrating the necessity for the introduction of generalization of the classic solution notion, is the following differential equation, representing in the simplified way the closed-loop control task with the discontinuous function of the feedback controller (e.g. in the classic problem of the minimal-time control):

$$ \dot{y}_{1} (t) = 1,\quad \quad y_{1} (0) = y_{01} $$

(84)

$$ \dot{y}_{2} (t) = \left\{ {\begin{array}{*{20}c} { - 1} & {\text{gdy}} & {y_{2} (t) > 0} \\ 1 & {\text{gdy}} & {y_{2} (t) \le 0} \\ \end{array} } \right.,\quad y_{2} (0) = y_{02} , $$

(85)

where $y_{01} ,y_{02} \in {\mathbb{R}}$. Its solutions end after reaching the axis $y_{1}$. If the solution could be extended, then as results from the formula (85), it would be “push” above the axis $y_{1}$ (because $\dot{y}_{2} (t) = 1$ when $y_{2} (t) \le 0)$, but immediately after its “release”, “pushed” in this axis again (since $\dot{y}_{2} (t) = - 1$ when $y_{2} (t) > 0)$, which negates the possibility of the existence of such a solution. It is worth observing that in the above example the function g is discontinuous on the axis $y_{1}$.

The concepts generalizing the notion of the classic solution for the needs of differential equations with a discontinuous right-hand side (when the function g defined in the formula (85) is discontinuous not only with respect to variable t) did not lead to a uniform approach. Currently the most frequently applied are solutions proposed by Caratheodory, Filippov and Krasovski. Before they have been defined, first the notions of “almost everywhere”, the absolutely continuous function, convex closed hull, and compact set will be recalled.

If the measure of the points of the set $A \subset {\mathbb{R}}$, in which some property is not fulfilled, equals zero, then one can say that this property occurs almost everywhere in A.

Now consider the function $x:\left[ {p,q} \right] \to {\mathbb{R}}$. It is called an absolutely continuous function if for every $\varepsilon > 0$ there exists $\delta > 0$ such that when $\left( {p_{1} ,q_{1} } \right)$, $\left( {p_{2} ,q_{2} } \right), \ldots ,\left( {p_{k} ,q_{k} } \right)$ are intervals included in the domain $\left[ {p,q} \right]$, separable and such that $\sum\nolimits_{i = 1}^{k} {\left( {q_{i} - p_{i} } \right)} \le \delta$, then $\sum\nolimits_{i = 1}^{k} {\left\| {x(q_{i} ) - x\left( {p_{i} } \right) \le \varepsilon } \right\|}$ is fulfilled. The absolutely continuous function is also continuous. Furthermore, it is differentiable almost everywhere in $\left[ {p,q} \right]$. These two properties particularly predispose the absolutely continuous function to be used for generalization of classic notion of the differential equations solution.

The convex closed hull of the set $C \subset {\mathbb{R}}^{n}$, denoted further as ${\text{conv}}(C)$, is called the smallest (in the sense of an inclusion relation) closed and convex set including the set C. For example, the convex closed hull of the two different points is a segment connecting them, while of three different points, a closed triangle, whose points are the apexes.

The subset of the space ${\mathbb{R}}$ is compact, if and only if, it is bounded and closed.

Now, the solutions of differential equations in the Caratheodory, Filippov, and Krasovski senses [11] will be defined. The function y, absolutely continuous on every compact subinterval of the set T, is a solution of the differential equations (82)–(83):

• in the Caratheodory sense (C-solution), if the Eq. (82) is fulfilled almost everywhere in T, and also the condition (83) is fulfilled;

• in the Filippov sense (F-solution), if

$$ \dot{y}(t) \in F[g]\left( {y(t),t} \right)\quad {\text{almost}}\;{\text{everywhere}}\;{\text{in}}\;T, $$

(86)

and also the condition (83) is fulfilled, whereas the Filippov operator F is defined as follows

$$ F[g]\left( {y(t),t} \right) = \bigcap\nolimits_{r > 0} {\bigcap\nolimits_{{\begin{array}{*{20}l} {Z \subset {\mathbb{R}}^{n} :} \hfill \\ {\mu (Z) = 0} \hfill \\ \end{array} }} {{\text{conv}}\left[ {g\left( {\left( {y(t) + rB} \right)\backslash Z,t} \right)} \right];} } $$

(87)

• in the Krasovski sense (K-solution), if

$$ \dot{y}(t) \in K[g]\left( {y(t),t} \right)\quad {\text{almost}}\;{\text{everywhere}}\;{\text{in}}\;T, $$

(88)

and also the condition (83) is fulfilled, whereas the Krasovski operator K is given by the formula

$$ K[g]\left( {y(t),t} \right) = \bigcap\nolimits_{r > 0} {{\text{conv}}\left[ {g\left( {\left( {y(t) + rB} \right),t} \right)} \right]} ; $$

(89)

where B denotes an open unit ball in the space ${\mathbb{R}}^{n}$ and μ is n-dimensional Lebesgue measure. C-, F- or K-solutions are unique, if each, respectively, C-, F- or K-solution is a function identically equal to it. The interpretation of these definitions presented below also provide practical motivations for the forms introduced above.

Thus, in the case of the C-solutions, the derivative $\dot{y}(t)$ is dependent (except the independent variable t) only on the current value of the solution y. This type of solution is in practice a mathematically uniform generalization of the classic solutions “joining”.

In the case of the K-solution, the values of the functions g are taken into account not only at the point $y(t)$, but at all points in the ball with the center at this point and positive, although, because of the set-product, of any small radius r. Furthermore, the convex closed hull supplements thus created a set of the function g values with intermediate points. Interpreting the above, one may state that the K-solution, allowing all points from the surrounding of the value $y(t)$, takes into account the measurements errors, unavailable in practice.

Finally, additionally introduced in the definition of the F-solution set-product implies that from the ball with center at point $y(t)$ and radius r, the zero-measure sets, irrelevant from the practical point of view, are eliminated.

Directly from the above definitions it results that:

• (A) every C-solution is a K-solution, because $g\left( {y(t),t} \right) \in K[g]\left( {y(t),t} \right)$;

• (B) every F-solution is K-solution, since $F[g]\left( {y(t),t} \right) \subset K[g]\left( {y(t),t} \right)$.

The uniqueness of K-solutions thus implies the uniqueness of C- and F-solutions.

Other relations between C-, F- and K-solutions are illustrated below by the example of the following differential equation:

$$ \dot{y}_{1} (t) = \left\{ {\begin{array}{*{20}c} 1 & {\text{when}} & {y_{2} (t) > 0} \\ { - 1} & {\text{when}} & {y_{2} (t) = 0} \\ 1 & {\text{when}} & {y_{2} (t) < 0} \\ \end{array} } \right.,\quad y_{1} (0) = y_{01} $$

(90)

$$ \dot{y}_{2} (t) = \left\{ {\begin{array}{*{20}c} { - 1} & {\text{when}} & {y_{2} (t) > 0} \\ 0 & {\text{when}} & {y_{2} (t) = 0} \\ 1 & {\text{when}} & {y_{2} (t) < 0} \\ \end{array} } \right.,\quad y_{2} (0) = y_{02} $$

(91)

where $y_{01}$, $y_{02} \in {\mathbb{R}}$ (Fig. 6).

Fig. 6

Solutions of differential equations (90)–(91)

First consider the case, when the initial state is included in the axis $y_{1}$. The function $\left[ {\begin{array}{*{20}c} {y_{01} - t} \\ 0 \\ \end{array} } \right]$ for $t \in \left[ {0,\infty } \right)$ is then a unique C-solution. Because for every $t \in \left[ {0,\infty } \right)$ and $y(t)$ contained in the axis $y_{1}$, the equality $F[g]\left( {y(t),t} \right) = \left\{ {\left[ {\begin{array}{*{20}c} 1 \\ v \\ \end{array} } \right]{\text{where}} \left| v \right| \le 1} \right\}$ is true, then the function $\left[ {\begin{array}{*{20}c} {y_{01} + t} \\ 0 \\ \end{array} } \right]$ for $t \in \left[ {0,\infty } \right)$ is then a unique F-solution. In turn, for every $t \in \left[ {0,\infty } \right)$ and $y(t)$ contained in the axis $x_{1}$, the set $K[g]\left( {y(t),t} \right)$ is a triangle with apexes $\left[ {\begin{array}{*{20}c} 1 \\ 1 \\ \end{array} } \right]$, $\left[ {\begin{array}{*{20}c} 1 \\ { - 1} \\ \end{array} } \right]$, $\left[ {\begin{array}{*{20}c} { - 1} \\ 0 \\ \end{array} } \right]$, therefore, the absolute continuous functions of the form $\left[ {\begin{array}{*{20}c} {y_{1} (t)} \\ 0 \\ \end{array} } \right]$, while $y_{1} (0) = y_{01}$ and $\left| {\dot{y}_{1} (t)} \right| \le 1$ almost everywhere in $\left[ {0,\infty } \right)$, are K-solutions.

In the second case, when the initial state is not included in the axis $y_{1}$, the function $\left[ {\begin{array}{*{20}c} {y_{01} + t} \\ {y_{02} - t \cdot {\text{sgn}}\left( {y_{02} } \right)} \\ \end{array} } \right]$ is, to the moment it crosses this axis, a unique C-, F- and K-solution. When it reaches the axis $y_{1}$, the solutions can be extended by the proper solutions described in the previous paragraph.

From the above example it results that in the general case there are no relations between C- and F-solutions, so continuing the comments formulated previously in the form of points (A)–(B), one can also add that:

• (C) C-solution need not be F-solution;

• (D) F-solution need not be C-solution.

Furthermore, K-solutions frequently rather constitute too rich, from a practical point of view, class.

Now suppose that the functions u and g are continuous, which assumption, as mentioned, is practically inherent to considerations of classic solutions. Because the right-hand side of the Eq. (82) is continuous, so the same is true for a derivative of such a solution, therefore $y \in {\mathcal{C}}^{1}$. The function of the class ${\mathcal{C}}^{1}$ on the compact interval is a Lipschitz function, so also absolutely continuous; therefore, the classic solution is then a C-solution. Due to point (A) it is also a K-solution. The above conclusions remain true also in the case when functions u and g have a finite number of discontinuities of the first type (the second of them with respect to the independent variable t), so in the case when the classic “joint” procedure can be used. In conclusion:

• (E) if the functions u and g are continuous or have a finite number of discontinuities of the first type (the second of them with respect to the independent variable t), then every classic solution is a C-solution.

Finally, in practice C- and in consequence K-solutions constitute a generalization of the classic solutions, also with the possibility of “joining”.

The relations between particular types of solutions have been synthetically expressed in the form of points (A)–(E). C-solutions are most frequently used in the case of discontinuities of the function g with respect to the independent variable t, F-solutions for discontinuities with respect to $x(t)$, although K-solutions generalizes the above two types, however often constitute too rich a class. The confusions to which the lack of a proper understanding and using of notions of solutions to differential equations with discontinuous right-hand side, were illustrated in the publications ([20], point 2.3.2; [15]).

In the first example of Subchapter 2 it was mentioned that in the hypothetical case $\widehat{M} = M$, trajectories have regular character and can be even described by classic solutions with “joining” or, more generally, C-solutions. If the parameter $\widehat{M}$ is overestimated, the over-regulations appear in the system; the trajectories are then C-solutions. And finally, when it is underestimated, then the robust control generates the sliding trajectories, described by F-solutions. In order to unify these cases, one can use the concept of K-solutions, more general then both C- and F-solutions. However, as mentioned, in many applications K-solutions constitute too rich a class, in the publications [13, 14] it was shown that in this case they are unique, and furthermore in the probabilistic approach, the family arrived in this way constitutes a stochastic process, which enables the use of advanced methodology available for this purpose.