Practical Stabilization for Uncertain Pseudo-Linear and Pseudo-Quadratic MIMO Systems ()
1. Introduction
The problem of practical stability and stabilization for linear and nonlinear systems subject to disturbances and parametric uncertainties together with an efficient robust control has been in the past [1-10] one of the most research topic and nowadays remains actual and significant [11-21].
Indeed there exist many controlled or not systems linear but with uncertain parameters, uncertain pseudo-linear and with bounded coefficients, uncertain pseudoquadratic and with bounded coefficients, having a bounded additional term, for which not always there exists an equilibrium state.
Regarding this, consider:
• the mechanical systems with not viscous friction and/ or with revolute joints (e.g. robots)• the electrical and/or electro-mechanical systems with ferromagnetic devices• many chemical, ecological, meteorological, biological and medical systemswith possible disturbances and reference signals which are non standard (not polynomial or cisoidal).
For the above significant systems, it is important to design a control law such that, in a finite time interval, the state evolution, for all the initial conditions belonging to a specified compact set, is bounded and such that the evolution of the output (also the error signal) converges, with assigned minimum velocity, to chosen maximum values, that are bounded and not necessarily null.
In this paper a systematic method, in a more general framework with respect to the ones proposed in literature (see e.g. [1-5,8-10,13-18]), for the analysis and for the practical stabilization of a significant class of MIMO uncertain pseudo-linear and pseudo-quadratic systems, with additional bounded nonlinearity and/or bounded disturbances, is considered. In detail, by using the concept of majorant system, via Lyapunov approach, new fundamental theorems, from which derive explicit formulas and efficient algorithms to design state feedback control laws, with a possible imperfect compensation of nonlinearities and disturbances, are stated. These results guarantee a specified convergence velocity of the linearized system of the majorant system and a desired steady-state output for generic uncertainties and/or generic bounded nonlinearities and/or bounded disturbances (see also [19-21]). Finally two significant examples of application, well showing the utility and the efficiency of the proposed results, are reported.
2. Problem Formulation and Preliminary Results
Consider the following class of uncertain quadratic multivariable systems
(1.1)
where:
is the output,
is the control input,
models possible external signals and/or particular nonlinearities of the system,
, with P a compact set of
, is the vector of the uncertain parameters,
and
are bounded matrices, continuous with respect to their arguments,
is a matrix continuous with respect to its arguments and with rank m.
Now suppose that there exist the nominal values
,
,
of
,
,
such that
,
are functions multilinear with respect to
and with respect to bounded function
where
![](https://www.scirp.org/html/4-2340056\1c5e149b-5173-4217-8ae6-b5f5995b6d68.jpg)
and
are hyper-rectanglesand such that
, where
is a constant.
Pose:
(1.2)
and denote with
,
, the matrices whose
rows are respectively the i-th rows of the m matrices ![](https://www.scirp.org/html/4-2340056\2644df2b-58e8-4db9-a071-f38793f91e7c.jpg)
Then, by controlling the system (1.1) with the following state feedback control law with a partial compensation
(1.3)
it is easy to verify that the closed-loop system turns out to be
(1.4)
To develop a practical stabilization method for the system (1.4), in a more systematic and general framework, which allows to calculate in a simple manner a control law that guarantees a specified convergence velocity, the following notations, definitions and preliminary results are provided.
(1.5)
where
is a symmetric and positive definite
matrix and
is a compact set.
Definition 1. Give the system (1.4) and a
symmetric matrix
. A first-order positive system
,
, where
and
such that
is said to be majorant system of the system (1.4).
Theorem 1. Consider the quadratic system
(1.6)
If
, it is (see Figure 1)
![](https://www.scirp.org/html/4-2340056\4eb799a6-f5fa-4d87-ad2c-23d4157ba96e.jpg)
Figure 1. Graphical representation of the system (1.6).
, (1.7)
where
, are the roots of the algebraic equation
. Moreover for
the practical convergence time
is given by (see Figure 2)
(1.8)
in which
is the time constant of the linearized of the system (1.6) and
is the upper bound of the convergence interval of
for
, i.e. of the system (1.6) in free evolution.
Proof. The proof of (1.7) easily follows from Figure 1. Instead (1.8) easily derives by noting that the solution of (1.6) for
is
. (1.9)
In Figure 2 the evolution of
as a function of
is reported. By analyzing Figure 2, note that for
it is
i.e.
.
Theorem 2. The solution of the Equation (1.6) with
is
(1.10)
where
, are the roots of the equation
.
Proof. From (1.6) it derives
(1.11)
from which (1.10) easily follows.
Lemma 1. If
is a symmetric and p.d. matrix,
is a symmetric matrix, continuous with respect to
, and
is continuous with respect to
, then
it is:
![](https://www.scirp.org/html/4-2340056\23594452-138b-41a1-81f2-a464a831fd38.jpg)
Figure 2. Factor γ as a function of the normalized initial condition.
(1.12)
(1.13)
Moreover, if
is linear with respect to x it is
(1.14)
More in general, if
is pseudo-linear with respect to x with bounded coefficients, i.e. if
where
are bounded, then
(1.15)
Proof. Note that, if
is a continuous function with respect to
and
are compact subsets of
, it is
,
. Moreover, since P is p.d., there exists a symmetric nonsingular matrix
such that
. Hence, by posing
, it is
(1.16)
and so (1.12) holds. Similarly
(1.17)
and hence (1.13). The inequalities (1.14) easily follow from the fact that, if
is linear with respect to
,
. The inequalities (1.15) analogously follow.
Remark 1. Clearly, if
and
are independent of
, (1.12) and (1.13) are valid with the equal sign. If
depends on x,
is quite difficult to compute because
has in general different points of relative maximum, of relative minimum and of “inflection”; moreover, the second and the third member of (1.12) (of (1.14) or of (1.15)) allow an easier computation of a lower bound on
proportional to
, as it will be shown later on. A similar talking is valid if
depends on
.
Lemma 2. Consider a p.d. matrix
and a matrix
with rank m. If
then the smallest
such that
where
is equal to ![](https://www.scirp.org/html/4-2340056\37c519fc-2de7-4830-8fb1-48634d2d1d33.jpg)
Proof. Since P is p.d., by posing
, where
is a symmetric nonsingular matrix such that
, then, in an equivalent way, the smallest
such that
is also the smallest
such that the matrix
is positive semidefinite, i.e. all its eigenvalues are non-negative. As
it is
By taking into account that, if
is a real
matrix with rank
, the matrix
has
null eigenvalues and
positive eigenvalues equal to the ones of
it follows that
and, hence, the proof.
Lemma 3. Let
be a symmetric p.d. matrix with its inverse
having unitary elements on the main diagonal. Then the hyper-rectangle of
externally tangent to the hyper-ellipse
is the hyper-quadrilateral (quadrangle if
, cube if
) having as origin the centre and with unitary half-side.
Proof. It is easy to prove that the point of contact of the hyper-line orthogonal to the versor
of
and tangent to the hyper-ellipse E is
. (1.18)
From this the proof easily follows.
The following significant and useful theorem is stated.
Theorem 3. Let
![](https://www.scirp.org/html/4-2340056\0cc0c18e-b97b-4d33-af77-d2792e63c20c.jpg)
be a matrix multilinearly depending on the parameters
![](https://www.scirp.org/html/4-2340056\8fed3310-3435-4315-a4ea-5151b1e15c7d.jpg)
and let
be a symmetric p.d. matrix. Then the minimum (maximum) of
, where
, is attained in one of the
vertices of
.
Proof. Note that for a constant
it is
Moreover, by taking into account that
, it turns out to be that
(1.19)
Therefore, said
the points of minimum of
, it is
(1.20)
From (1.20) the proof easily follows.
3. Main Result
Now the first main result, concerning the analysis of stability, can be stated.
Theorem 4. Give a symmetric p.d. matrix
. Then a majorant system of the system (1.4) is
(1.21)
in which:
(1.22)
where
and
are the sets of vertices of the hyperrectangle
and of the hyper-rectangle
respectively, and
is the set of vertices of the hyper-rectangle of
externally tangent to the hyper-ellipse
.
Proof. By choosing as “Lyapunov function” the quadratic form
, for x belonging to a generic hyper-circumference
, it is
(1.23)
The proof easily follows from (1.23), Lemmas 1, 2 and Theorem 3.
The second main result, concerning the synthesis of the stabilizing control law, follows from Theorem 4.
Theorem 5.
Let
be the characteristic polynomial of the low-pass Butterworth filter of order
with cutoff frequency
.
If in (1.3) it is posed
(1.24)
in which I is the identity matrix of order m, then a majorant system of the system (1.4) with respect to the norm
, with
, where
(1.25)
being
,
, the k-th root of
for
, is
(1.26)
in which:
(1.27)
(1.28)
(1.29)
Proof. By making the change of variable
, it is easy to prove that the system (1.4) becomes
(1.30)
Moreover, note that the Butterworth eigenvalues
have unitary magnitudes; hence all the main diagonal elements of the matrix
are unitary. From this consideration, from (1.30), from Lemmas 2, 3 and Theorem 4, the proof easily follows.
Theorem 6. For
, the parameters
and
of the majorant system (1.26) turn out to be
(1.31)
(1.32)
Proof. It is easy to verify that for
the matrix
is the limit of the eigenvectors matrix of the matrix
(see (1.27)); hence it is that
,
, where
. Therefore
(1.33)
hence (1.31).
Moreover it is easy to verify that
(1.34)
From (1.28) and (1.34) the expression (1.32) easily follows.
Remark 2. If
it is easy to prove that
(1.35)
From Theorems 5 and 6 the following result derives.
Theorem 7. Consider the system (1.4) with
provided from (1.24). If the design parameter a is big enough, from a practical point of view, the time constant
of the linearized of the majorant system (1.26) is inverse proportional to a and it coincides with the maximum time constant of the linearized of the system (1.4). More in detail, if a is large enough it turns out to be
(1.36)
moreover, if a is sufficiently large, for t large enough it is
(1.37)
or, more in general, it is
. (1.38)
Proof. (1.36) easily follows from (1.26), (1.31) and by noting that
. (1.39)
The inequality (1.37) follows from (1.7), from the fact that if a is sufficiently large then
from the second of (1.26) and from (1.29) and (1.31).
(1.38) analogously follows by taking into account that
(1.40)
4. Examples
The following examples show the utility and the efficiency of the results stated in the previous sections.
Example 1. Consider the pseudo-linear uncertain system
, (1.41)
where
and
. By posing
and by applying Theorem 5, the majorant system of the system (1.41) controlled with the control law
turns out to be
(1.42)
In Figure 3 the value of
for
is reported. It is significant to note that for
it is
, i.e.
unless 5%, in accord with Theorem 7. For
it is
hence
. Moreover, being
, it is
and
. Therefore, at “steady state”,
it is:
and
.
Example 2. Consider the system of Figure 4 described by the equation
(1.43)
in which
are the disturbance actions due to several causes included the slope of the road,
,
,
and
![](https://www.scirp.org/html/4-2340056\6eb1b44f-876a-4d68-abd7-89f74018ee4c.jpg)
By posing
,
,
,
and by applying Theorem 5 the majorant
system of (1.43), controlled by using the control law
(1.44)
turns out to be
(1.45)
In Figure 5 the value of
for
is reported. It is significant to note that for
it turns out to be
, i.e.
unless 10%, in accord with Theorem 7. For
it is
; hence
. Moreover it is
,
773.0. Therefore, at “steady state”,
,
,
, it is:
![](https://www.scirp.org/html/4-2340056\aee9574d-0202-4e3a-b470-1669fff2ee48.jpg)
Figure 6 shows the values of
and of
, obtained for
,
,
,
,
,
and
square waves of amplitude 1 and frequency 1.2 Hz,
,
.
This figure highlights that the proposed stabilization method is little conservative, as it can be easily verified by simulating the stabilized system for several initial conditions and numerous values of the parameters.
5. Conclusions
In this paper the problem of analysis and practical
![](https://www.scirp.org/html/4-2340056\9b423e33-3d20-48db-ab99-ea4f1dd1452a.jpg)
Figure 4. Vehicles to be kept still at a distance ds.
![](https://www.scirp.org/html/4-2340056\44e889ff-4368-42a2-af5c-f4ebae5b05ff.jpg)
Figure 6. Possible time histories of
and of
.
stabilization of a significant class of MIMO nonlinear systems subject to parametric uncertainties, including linear and quadratic ones with an additional bounded nonlinearities and/or disturbances, has been approached. By using the concept of majorant system and via Lyapunov approach, new useful results, explicit formulas and efficient algorithms for designing state feedback control laws, with a possible imperfect compensation of nonlinearities and disturbances, have been stated. These results have been proved that guarantee a specified convergence velocity of the linearized of the majorant system and a desired steady-state output for generic uncertainties and/or nonlinearities and/or bounded disturbances.
The utility and the efficiency of the these results have been shown with two illustrative example.
The presented results can be used to establish further new useful theorems for the tracking of trajectories for relevant MIMO systems, like e.g. the robots.
In this direction the research of the author is going on.