Analysis and design of nonlinear dynamical systems using mathematica

Nonlinear

Analysis,

Theory,

Methods

Pergamon PII: SO362-546X(96)00350-1

ANALYSIS

AND

& Applicotionr, Vol. 30, No. 6, pp. 379553805, 1997 Pmt. 2nd World Congress of Nonlinear Analysts Q 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/97 S 17.00 + 0.00

DESIGN OF NONLINEAR DYNAMICAL USING MATHEMATICAB

SYSTEMS

JESUSRODRIGUEZ-MILLAN Universidad

de Los Andes, Facultad de hgenier$ Apartado NP 11, La Hechicera, Men ph: 5874402987, Fax: 5%74402872, Email:

enb de Sistemas

de Control

[email protected]

Key words and phrases: Symbolic and graphical corn utation, Mathematical, systems, nonlinear oscillations, PoincarCAndronovI-? opf brfurcations, . Lyapunov nonlinear control, Jacobian and extended linearization, state feedback controllers, pID controllers, lead-lag compensators.

Analysis of dynamical stability, dissipativity, Luenberger observers,

1. INTRODUCTION

The desire and/or the need of automating long, tedious, complex and error prone sequencesof mathematical symbolical computations has been a dream as old as mathematics itself. The realization of this dream is still quite far from complete, but the spring up of symbolic computation systems during the last thirty years seems to be a step forward in the right direction. Symbolic computation is not a panacea. However, it is not hard to imagine that its development, synergetically integrated with both graphical and traditional numerical computation techniques, will have an enormous impact on the evolution of mathematics and all other physical and mathematical sciences,in the near future. In this paper we will illustrate how symbolic computation tools can be successfully used to approach the solution of some elementary problems in the analysis of nonlinear dynamical systems,and the design of nonlinear controllers and observers for nonlinear controlled dynamical systems. The paper is organized as follows: section 2 covers some general aspects of symbolic computation and its application to nonlinear dynamics. Section 3 surveys the development of some symbolic computation tools for the analysis and design of nonlinear dynamical systems.In sections4 and 5 we present a survey of some of our own work on symbolic computation for the analysis of second order nonlinear dynamical systemsand the design of nonlinear controlled dynamical systems. We illustrate these developments through appropriate casestudy examples. 2. GENERALITIES

ABOUT

SYMBOLIC

COMPUTATION

The origin of symbolic computation, i.e., the use of digital computers to perform mathematical symbolic calculations according to previously defined analytical algorithms, can be traced back to the beginning of the decade of the sixties [l], when the introduction of LISP (1960-1961) allowed to write the first computer programs to automate the symbolic calculation of derivatives and indefinite integrals. The evolution of symbolic computation during these thirty years can be very schematically divided into three periods: the experimental period of research and implementation (1961-1971), the initial phase of its use as a production tool (1971-1981), and the period of diffusion and commercialization, from 1981 on. As a whole, the period 1961-1981 conformed a closed period in the evolution of symbolic computation, because only those researchers with accessto enormous computation facilities were able to develop new symbolic computation systems (SCS) or benefit from the already existing ones. The real expansion of symbolic computation as a tool for scientific research and education began in 1979, with the introduction of muMath: the first SCS for IBM PC environments. However, it was not until the middle of the decade of the eighties when the SCS for individual workstations and IBM PC 3795

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compatible computers became powerful enough as to allow their use for solving difficult mathematical problems in these environments. Thus, the application of symbolic computation in the big majority of scientific disciplines is not older than ten years, and its expansion for teaching purposes is still more recent. Looking towards the future, and taking into account that processes automation (no matter whether we mean by a process the optimization of an industrial complex or the proof that a given polynomial belong to a particular ideal) is only possible when it can rely upon a supporting constructive algorithm, the very existence of symbolic computation possesses a big challenge to contemporary mathematics: how to solve problems through constructive algorithms that work out their results in a finite number of steps, and whose complexities do not increase exponentially fast with respect to neither the order of the models nor the size of their internal coefficients. In what to nonlinear dynamics is concerned [2], the traditional numerical computation techniques had basically transformed computers into formidable simulation tools, that allow to track the temporal evolution of particular trajectories as precisely as needed. This approach, as powerful as it has proved to be, is not well tailored for the study of qualitative aspects of nonlinear dynamics, such as, for example, the dependence of the asymptotic properties of solutions with respect to initial conditions and parameters, the effects that parameters’ perturbations have on the global structure of the flow, or the carrying out of symbolical experiments. The difficulties associated to the symbolic analysts of nonlinear dynamical systems [2] might be classified into four groups, according to whether they come out from: (ij the lack of a deep enough level of theoretical understanding, (ii) the lack of constructive algorithms to solve particular problem, (iii) technical difficulties to implement already known constructive algorithms, and (iv) the length, complexity and tediousness of the symbolical calculations involved. The last group of difficulties might be overcome using already existing SCS, such as Mathematics, Maple, Macsyma, etc., to automate long, tiring and error prone sequences of standard symbolic manipulations. The surmounting of the difficulties of the third group will depend, however, not only on the increasing speed,reliability and internal programming facilities of future SCS, but on the development of new and more efficient symbolic calculation algorithms to solve basic mathematical problems: the calculation of the zeros of functions, the solution of linear and nonlinear systems of simultaneous equations, the resolution of integrals, and the calculation of maxima and minima of arbitrary functions, for instance. Symbolic computation is still in a period of very fast evolution, both from the mathematical and computational points of view, and future SCS will surely play a very important role in the symbolic study of nonlinear dynamics. This notwithstanding, it would be very naive to think of SCSas a definite tool to deal with the qualitative theory of dynamical systems. A more probable future scenario will be one, where nonlinear analysts will approach problems using integrated symbolic-graphical-numerical computational tools. 3. SYMBOLICS

FOR NONLINEAR

CONTROLLED

DYNAMICAL

SYSTEMS:

A BRIEF

SURVEY

In this section we survey the development of some symbolic computation tools (SCT) for the analysis and design of nonlinear controlled dynamical systems (NLCDS). For a deeper survey, with emphasison the mathematical foundation of the SCT mentioned below, we refer the reader to our previous work 121 The first applications of symbolic computation to the analysis and design of NLCDS date from the early eighties: In 1984Cesaro and Marino reported their first experiences with REDUCE on the controllability analysis of elastic robots [3] and the stabilization of an electrical power system [4]. The first SCT for the design of nonlinear observer, using Macsyma as symbolic computation platform to implement the two steps design algorithm of Keller, was reported by Bar, Fritz and Zeitz [5] in 1987. Also in 1987, Akhrif and Blankenship introduced the Macsyma based SCT CONDENS, containing modules for linearization and inversion of NLCDS. In 1989Birk and Zeitz [6] introduced the first version of MACNOM (MACsyma program for

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NONlinear systems), emphasizing the design of extended Luenberger observers according to the approach of Zeits 171and Birk and Zeitz [B]. MACNOM also possesses modules for Lyapunov stability, transforming NLCDS into the observer and observability canonical forms, and the design of Kalman filters. A new expanded version of MACNON was described in [9] in 1991, incorporating a new submodule for the design of state feedback controllers, transforming NLCDS into the controller and controllability canonical forms. In 1993 RothfuiJ [lo] reported the addition of a new submodule to MACNOM to perform stability studies according to the first and second methods of Lyapunov, or the center manifold method for NLCDS with non hyperbolic equilibrium points. The SCT developed by Ogunye and Penlidis for the analysis and design of controlled systems can be divided in two groups: a first one for the study of controllability, observability [II] and balanced realizations 1121,and a second one 113,141for the analysis of time-invariant linear controlled dynamical systems (LCD’S). In [ll] they use the full rank condition of the controllability (observability) grammian to check the controllability (observability) of the LCDS, instead of the usual Kalman rank conditions. The SCT for the analysis of LCDS include modules for the: calculation of transition matrices, discretization of LCDS described in state variables, solution of discrete-time LCDS given in state variables, transformation of transfer functions into the observer, controller, and Jordan canonical forms, etc. In 1991 Bram de Jager used Maple to develop ZeroDyn [15], his first SCT for the study the zero dynamics in NLCDS that are linear with respect to the control signal. Later on van Essen and de Jager developed NonLinCon [16,17], an improved version of ZeroDyn, to approach the study of zero dynamics and the solution of the exact linearization problem. In [18,19] de Jager discoursesupon the present limitations of Maple regarding the development of SCT for nonlinear control, and proposes the development of hybrid or symbolic-numerical computation tools to deal with nonlinear control problems. The present status of the work of Blankenship et al on the development of SCT for modeling multibody dynamical systems and designing nonlinear and adaptive control laws based on dynamic inversion methods has recently been summarize in [201. The development of SCT based on algebraic methods and their application to nonlinear control problems is well represented by the work of Forsman and other members of the Swedish school of control. Forsman has reported the application of Grobner basesto the construction of Lyapunov functions [21], nonlinear robustness theory [22], the symbolical solution of the algebraic Riccati equation, etc. In [23] Forsman introduced the Maple SCT POLYCOM which is oriented towards the analysis of NLCDS where all nonlinearities are polynomial or rational functions. Svensson [24] has also reported the development of Maple SCT for exact linearization, analysis of NLCDS using Volterra series, stabilizing feedback and nonlinear optimal feedback. 4. NLCONTROL:

A SYMBOLIC

COMPUTATION

TOOL

FOR NONLINEAR

CONTROL

USING

MATHEMATICA@

After gaining some experience with Mathematical on the developing of SCT to study nonlinear oscillations in parameter depending second order nonlinear dynamical systems, seenext section of this paper, the author and his coworkers started a program to develop NLControl, a SCT for nonlinear controllers and observers design. In this section we describe the present status of our work, emphasizing those aspects not widely available elsewhere. A detailed teaching oriented presentation of the algorithms and techniques supporting NLControl is reported in [25]. In the sequel we will consider smooth enough single-input single-output nth order NLCDS described in state variables, i.e., x’ = f(x,u), y = h(x),

x E 31”, u E R, y E R, f and h smooth enough,

(4.11

that must be asymptotically stabilized to a desired operating point (U, X(U), Y(U)). The linearization of the state equation and the output equation in (4.11 around W, X(U), Y(U)) are

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given by x’ = Dxf(U,X(U))x + Duf(U,XfU))u = AW)x + B(U)u y = DxhKJ,X(U))x = C(IJ)x

(4.21 (4.3)

respectively. Until now we have developed a set of SCT to approach the solution of this problem using the Jacobian linearization method (JLM) and the extended linearization method (ELM) to synthesize the required stabilizing linear and nonlinear control laws, respectively. The present version of NLControl consists of four modules: NLFeedback, NLObserver 1261, NLPID [27], and NLLead-Lag, to design state feedback controllers for NLCDS with completely controllable linearization, Luenberger observers for NLCDS with completely observable linearization, PID controllers and lead-lag compensators, respectively. Figure 1 shows the general flow diagram of NLControl, where the four modules we mentioned above have been identified by a, B, y and S, respectively.

I

Translation

Figure

to the origin

1. General

and linearization

flow diagram

of NLControl

As it ls suggested in the flow diagram above, the design of nonlinear controllers and observers using the ELM 128) goes in two steps: Firstly, we design a linear controller or observer using the JLM to satisfy a set of design specifications, and secondly we calculate a nonlinear extension of the previously synthesized linear controller or observer. The algorithms supporting the modules NLObserver and NLPID have already been described in 1261and [271, respectively. In the sequel we describe the structure and internal algorithms of the module NLFeedback. The’ module NLLead-Lag will be the subject of a forthcoming paper. It is well known that the complete controllability of the pair of matrices {A(U), B(U)} assures

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such that the roots of the characteristic

x’ = [A(U) - BW)KW)lx

(4.4)

can be arbitrarily placed on the complex plane. Thus, the complete controllability of (A(U), B(U)} allows to asymptotically stabilize the closed loop control system (4.4) according to any desired convergence pattern. Furthermore, (4.4) coincides with the linearization of the closed loop NLCDS x’ = f(x,u), u = - KW)xW,

(4.5)

and therefore, after the Hartman-Grobman theorem, the local dynamic of (4.5) in a small enough neighborhood of the origin is generically homeomorphically equivalent to the global dynamic of (4.4). Thus, the linear state feedback control law u = - K(U)x also generically asymptotically stabilizes the NLCDS (4.1) around the origin. To try to increase the domain of attraction of the origin in (4.5) the ELM prescribes [281 to search for a nonlinear extension u = -k(x(t)) of the previously designed linear control law u = - KfIJ)x, in the sense that the linearization of the closed loop NLCDS x’ = f(x,u), u = - k(x(t)), coincides boundary

with (4.4). Thus, the nonlinear value problem D,kfX(U))

U = - k(XtU)),

control law u = -k(x(t)) = K(U), k(X(U))

(4.6) must be the solution

= -U

of the (4.7)

to be a nonlinear extension of the linear control law u = - K(U)x in the required sense. Figure 2 shows the internal flow diagram of the NLFeedback module. a 0

1

Construct

x’ = IA(U)

- B(KJ)K(U)lx

and calculate

its characteristic

polynomial

I Use the design

Solve PDE @k/&1(

specifications

to calculate

the desired characteristic polynomial I Linear state fefdback control law: u = - KOx I X(U)) = K(U), Lo<(U) = -U, and get the nonlinear control law: u = - KOX 1

to the original

Return

coordinates

I

Figure

2. Internal

flow diagram

of the module

-back.

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For the class of NLCDS we are considering family of ordinary differential equations E

of Nonlinear

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in this paper, problem (4.7) is equivalent

[X(U)] = K,(U), k(X(U))

to the

(4.8)

= - U, 1 2 i I n.

System (4.8) admits a symbolical recursive solution in n steps, and therefore the extended nonlinear control law can be explicitly calculated. The n steps algorithm conforms the kernel of NLFeedback, and goes as follows: ak(x) Step 1. Solve ax,

I X(U)

k(x) = k&(U),

Step 2. wve

ak(x) K

= KI(U) to get (4.9)

x2, . . . , xn) +

I

x(u) = K2(U) to calculate k(XI(u),

x2, . . . , x,). You will get:

x2

k(X, 0,

x2, . . . , x,) = k(X,(U),

X,(u)),

. . . , x,) +

[K2(U)]

X2W)=q

d=,

x2 I U) (4.10)

dqdo2. -x2i)x,i,

EPI X1,2(U)=o1,2

ak(x) = KS(U) Step 3. Solve ax3 I X(U)

to calculate k(X1(U),X@J)),

ak(x) = K,(U) Step n. Solve %I I X(v) analytical expression for k(x):

to calculate k(X,(U),X2(U)),...,X,-,(U)),xn)

. . . I xn).

xi

n k(x) = k(Xl(u), X,(U), ... , Xn-l(v), X,(U))) + c i=l

+ $

z

n+l

+ (-1)

xj~jxi~J

[FIxi

jcu)=oi I

and get the final

[Ki(U)]Xi(U),o Xi (I

Fdoj I

)

+ ---

aK1(U) mcn”‘m(l 1Xn~,~~G-Jbn~..*

i

doi

+

dq...dq,,

(4.11)

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where koC~WM~KJL. ..,Xnl(U),Xn(U)) = - U. As we showed above, it is in principle possible to recursively solve the boundary value problem (4.7) to calculate the extended nonlinear control law u = - k(x(t)). However, from the practical point of view, the actual capability of NLFeedback to calculate u = - k(x(t)) depends on the ability of Mathematical to solve the integrals in (4.11). Thus, the very eventual limitation of NLFeedback on performing its task is not t:he order of the NLCDS (4.1), but the actual explicit calculation of formula (4.11). The use of NLFeedback and NLObserver to calculate nonlinear controllers and observers for NLCDS of different orders is illustrated in 1251. The already known limitations of presently available symbolic computation systems to simplify complicated mathematical expression is also present in NLFeedback: the calculated nonlinear control law for a system of four identical non interacting tanks in cascade is one page and a half long 1251. It is frequently possible to simplify the calculation of the nonlinear control law u = - k(x(t)) by hand, if we integrate the n ordinary differential equations in (4.8) according not to the natural sequence 1,2,3, . .. , but to a different guessed sequence 3,5, 2,1, .... for instance. To profit from this guessing ability of our brain, NLFeedback allows the user to solve (4.8) according to either the sequence of the natural numbers or an arbitrary prescribed sequence. An extension of the modules NLFeedback (NLObserver) to cover the case of NLCDS with stabilizable (detectable) linearization around the desired operating point is nearly ready, and will be describe in a future paper. A new version of the module NLLead-Lag is already in progress as well. 5. SOME SYMBOLIC SECOND ORDER

COMPUTATION NONLINEAR

TOOLS DYNAMICAL

FOR THE ANALYSIS SYSTEMS USING

OF PARAMETRIZED MATHEMATICAQ

In this section we will briefly survey our first experiences using Mathematics@ to develop SCT to detect and classify nontrivial periodic orbits in parametrized second order nonlinear dynamical systems (NLDS). This program started in 1992 and its present state of development could be summarized into four main complementary SCT, that we have called HartGrob, PAHopf, PICBox [29] and ProgSim, for reference purposes. 5.1. HARTGROB

The development dynamics around systems, i.e.,

of HartGrob, a symbolic graphical computation tool to study the local the equilibrium points in biparametric second order nonlinear dynamical

x’ = f(x, h), x

l

%*, h E R*, f smooth enough

(5.1)

began in 1992 [30], and has been continuously improved since then. From the internal calculation point of view HartGrob is a Mathematics@ based SCT, but the results of the symbolic calculations are used to construct the so called topological classification maps of the equilibrium points, that exhaustively describe the topological structures of the equilibrium points for all possible values of the parameters on the parameters space.The internal structure of Ha&rob is described in the flow diagram of Figure 3. The key point on constructing the topological classification map of an equilibrium point X(h) of (5.1) is the calculation of the zero and infinity level curves of the trace, the determinant and the discriminant of the linearization of (5.1) around X(h), becausethese curves divide the space of parameters into a set of disjoint compartments that describe the different topologies that X(h) might adopt as h assumes different values on the space of parameters. The topological classification criterion is based on the signs of trace, the determinant and the discriminant [31]. The name Ha&rob was adopted after the Hartman-Grobman theorem.

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5.2. PAHOPF

PAHopf is a symbolic graphical computation tool oriented towards the detection and classification of Poincare-Andronov-Hopf (PAH) bifurcations in NLDS like (5.1). From the conceptual point of view, PAHopf was designed to fulfill two purposes: to detect PAH bifurcations by checking the transversality condition of PAH bifurcation theorem, and to classify the already detected PAH bifurcations either by the integral averaging method [321 or the Poincare constant method [33]. The internal organization of PAHopf is described in the flow diagram of Figure 3. The results generated by PAHopf are resumed in a classification map on the space of parameters that described the subsets of parameters associated to supercritical or subcritical PAH bifurcations.

Zero and infinity

level curves of Det[A(h)] I Discriminant of A(L): Dis[A(h)l I I Zero and infinity level curves of Dis[A(h)l I Plot Zero and infinity level curves of Tr[A(L), DetIA0.)1, I

I

DidAWl

I

I

Topological

classification

map on the space of parameters I

Figure 3. Internal structure of the symbolic graphical computation

I

tool HartGrob.

The construction of the classification map is done in two main steps: firstly we calculate the stability index associated to the bifurcating family of nontrivial periodic orbits, which in general is a function of the parameters of system (5.11, and secondly we calculate the zero and infinity level curves of the stability index to determine the subsets of parameters associated to supercritical PAH bifurcations (negative sign) and subcritical PAH bifurcations (positive sign).

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Zero and infinity

Figure 4. Internal structure

Analysts

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averaging I

method

level curves

of the symbolic graphical computation

tool PAHopf.

5.3. PICE!oX

PICBox is a symbolic computation tool whose main purpose is the construction of positively invariant compact boxes to trap all the trajectories with unbounded initial conditions in dissipative second order NLDS. The positively invariant compact boxes are constructed using a Lyapunov function whose orbital derivative is bounded by a bounding function that can be represented as the addition of two scalar funrtions. The internal algorithms supporting PICBox have been described in detail in [291. One of the main applications of PICBox is the detection of persistent families of nontrivial periodic orbits under big perturbations of the parameters, through arguments based on dissipativity and the PoincarCBendixon theorem. 5.4. PROGSIM

ProgSim is a programmable simulator that allows to draw an arbitrary programmed sequence of phase diagrams, or temporal simulations, as the parameters of system (5.1) evolve along an arbitrary curve, or a given sequence of points, on the topological classification map generated by HartGrob. Amongst the possible applications of ProgSim we could mention: (i) experimental studies of the persistence of nontrivial periodic orbits for big ranges of parameters values, (ii) the experimental classification of PAH bifurcations when the analytical methods fail, (iii) experimental detection of homoclinic bifurcations, (iv) experimental studies of the global dependence of global dynamics on the parameters values. The four symbolic graphical computation tools described above can be used in an integrated way, Figure 5, to obtain additional informatio’n about the global structure of the set of trajectories of a particular NLDS. The PoincarCBendixon block in Figure 5 is not a XT, but a conceptual block to represent the use of the Poincar&Bendixon theorem and the information supplied by Ha&rob, PAHopf and PICBox to detect the existence of nontrivial periodic orbits. The proposed integrated approach was used in [:29] to prove the persistence of attracting nontrivial

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periodic orbits for big ranges of the parameters in Fitzhugh equations, and the coexistence of least two nontrivial periodic orbits when subcritical PAH bifurcations occur.

at

(Begin)

Figure

5. Interconnection

of HartGrob,

PAHopf,

ProgSim

and PICBox

ACKNOWLEDGMENTS

A big part of this paper is based on the work I did during my stage at the Automation Institute of the Hungarian Academy of Science (SZTAKII, in Budapest, between March and April of this year. I would like to express my deep feelings of gratitude to Prof. Jozsef Bokor for his generous invitation and support. The presentation of this work at the WCNA’ 96 was partially financed by the Merida State Government through Fundacite-Merida, and the Research Council of Los Andes University. REFERENCES 1.

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f

f

B