Continuous Piecewise Linear Control for Nonlinear Systems: The
Parallel Model Technique
Andr´es G. Garc´ıa Universidad Nacional del Sur Dto. de Ing. El´ectrica y de Computadoras Instituto de Investigaci´on en Ing. El´ectrica
(IIIE, UNS-CONICET) Av. Alem 1253, B. Blanca
Argentina
Osvaldo E. Agamennoni Universidad Nacional del Sur Dto. de Ing. El´ectrica y de Computadoras Instituto de Investigaci´on en Ing. El´ectrica
(IIIE, UNS-CONICET) Av. Alem 1253, B. Blanca
Argentina
Abstract: This paper describes a systematic technique for obtaining controllers for Nonlinear Systems using a
Continuous Piecewise Linear Approximation (CPWL) of the given Nonlinear vector field. The method proposes the use of a CPWL approximation of the Nonlinear System and then a theory is developed to show that the stabi-lization of the CPWL aproximation ODE yields stability for the Nonlinear ODE. An example is presented in order to show the capabilities of this idea but also the practical applicability. Finally some conclusions and future work are depicted.
Key–Words: Nonlinear ODE’s, Continuous Piecewise Linear ODE’s, Control Systems.
1
Introduction
Nonlinear Controller design is a very involved issue into the control community. While for linear systems there exists a wide range of methodologies from anal-ysis to readily implementable strategies (see [1], [2] and [4]), for Nonlinear dynamics only a few method-ologies are known to be effective and usually en-counter in practice hard problems regarding time con-suming or demanding tremendous amounts of capa-bilities from computer resources (see [6],[7] and [8]). Since Control problems are not more than a pa-rameterized Initial Value Problem (IVP) for Ordinary Differential Equations (ODE’s), for designing control strategies, is natural to resort primarily to techniques able to solve -or approximate- such a problems.
One methodology which has shown to be effec-tive either from numerical issues or theoretical anal-ysis is the use of Continuous Piecewise Linear Ap-proximations (CPWL), this idea was early studied by Sacks in a qualitative fashion (see [14]) and later ex-tended to approximate solutions by Girard, De Feo, Storace and Johansson (see [13], [19] and [11]).
Is worth noticing that two main research streams are related to Continuous Piecewise Linear (CPWL) ODE’s:
◦ Dynamic Systems which are written as CPWL ODE’s.
◦ As approximation to Nonlinear Continuous ODE’s.
First item was pioneered studied and defined by Chua in the early’s 700s (see [16] and [17]). This work gave origin to a Canonical Representation for a CPWL Basis able to represent any CPWL vector field developed by Juli´an (see [9]). Using this basis a reli-ably toolbox written in Matlab code were developed in order to approximate a given vector field with some desired degree of error (see [10]), on the other hand with a different procedure than a CPWL basis the Phd thesis of Johansson also leads a Matlab toolbox (see [11]).
Second item was conducted by Storace and De Feo who made several numerical experiments to in-vestigate topological properties of Nonlinear ODE’s of low dimensions. However, this work is not prov-ing in rigor that the properties of the given the Non-linear ODE and the CPWL approximation are shared by both systems, they only present extensive simula-tions and Continuation numerical packages to show this idea (see [15]).
One attempt to overcome this inconvenient pro-viding a way to decide if both (Nonlinear ODe and its CPWL approximation) share properties, are the works in [13] and [18] where Dynamic Error Bounds are de-rived. The former paper proved that the trajectories of the Nonlinear ODE and the trajectories of the approx-imation CPWL differs in norm around the order of the grid size -see [9] for a precise definition of gride size, while the second paper proved that the dynamics of
the error bound introduced in that paper is the same as the CPWL approximation ODE.
As is evident, the result in [18] when applied to a system with a stable CPWL approximation will lead the conclusion that the trajectories of both systems are stable and this precisely the topic of the present paper, the concept of ”Parallel Model”, that means, given a Nonlinear control system to the type x(t˙ = f(x) + B ·u and a CPWL approximation of f(x), fCP W L(x), then a CPWL controlleruCP W L(x) sta-bilizingxCP W L˙ (t) =fCP W L(xCP W L)+B·uCP W L is also stabilizingx(t˙ =f(x) +B·u.
A remarkable point here is the fast implementa-tion of CPWL funcimplementa-tions with microelectronics sug-gests that the future implementation of electronic con-troller could be appealing via CPWL vector fields which approximate any nonlinear dynamical system with some degree of accuracy, see [20]
This paper is organized as follows: Section 2 in-troduces precise definitions for the kind of Nonlin-ear systems considered in this paper and the goal ad-dressed ,Section 2.1 presents the formulation of an er-ror bound able to ensure stability for both systems, Section 3 provides an analysis of the asymptotic prop-erties of the approximate and real systems, Section 4 shows how the developed theory works in practice ap-plicability in a real case and finally Section 5 depict some conclusions and future directions for research.
2
ABOUT THE PROBLEM
CON-SIDERED
A Nonlinear Control System is a set of ODEs to the form X(t) =˙ f(X, u) where X ∈ <n and with X ∈ <m. In this context, one important aim is to determine a suitable controller u endowing the sys-tem with some desired properties (stability, asymp-totic stability, robustness, performance, etc).
As is very well known this aim is very difficult in practice for general cases with any nonlinear control system as exposed above, being available only a few partial results for special cases.
The idea in what follows is to produce an approx-imation of the Nonlinear Vector Field with a CPWL one, moving the task of designing the controller to the side of the approximate vector field and keeping track some error bound.
Notice that the design procedure for the obtained CPWL ODE can be carried out with the available techniques for designing of controllers for Piecewise Linear ODE’s in general as the one in [11] or [24]. However, the theory that it will be developed in the present paper will give support to the techniques in
[11].
In this paper we aim to develop a controlleru(t) for the following class of Nonlinear Systems:
˙
X(t) =f(X) +B·u(t), (1) whereX∈ <n,f :<n → <nis a smooth vector field andBis a constant matrixB ∈ <nm. The point is to approximate the Nonlinear vector fieldfby using a CPWL one as follows:
˙
XCP W L=fCP W L(XCP W L) +B·u(t), (2) whereX˜ are the approximate trajectories toXin (1). Incidentally notice that the same controlleru(t) is applied to both systems, moreover, if we define the control vectoru(t) to be Continuous Piecewise Lin-ear1with the same partition as forf(X), we have:
˙ X=f(X) +B·uCP W L ˙ XCP W L =fCP W L(XCP W L) +B·uCP W L uCP W L=Aiu·XCP W L+Bui (3) where the notation Ai andBi was introduced in [18] indicating the matrices forming the linear approx-imation into the simplexith. This is the reason why this technique will be called Parallel Model, since we need the trajectories of the CPWL model in or-der to be applied into uCP W L to effectively control
˙
X =f(X) +B·uCP W L
2.1 CPWL Approximations for Nonlinear Control Systems
The class of Nonlinear control systems defined in (1) can be described in a formal way as follows:
˙ X(t) =f(X) +B·uCP W L f(X) = [f1(X)· · ·fn(X)]T fi∈U ⊂ <n→ < i= 1, . . . , n X= [X1· · ·Xn]T,
where T stands for transpose. Let us considerU divided inrsimplices using a boundary configuration H as described in [9] with the CPWL approximation fCP W L in the simplexithto the given vector fieldf denoted by: ( fi CP W L(X) = [f1iCP W L(X)· · ·fnCP W Li (X)]T fi CP W L(X) =Ai·X+Bi 1u(t)
could be Piecewise Linear Discontinuous as discussed in [11], pp.98
In what follows we use the idea presented in [9],pp. 100, where the maximum norm was used in order to characterize the size of the error:
|f1(X)−f1iCP W L(X)| ≤ε1 |f2(X)−f2iCP W L(X)| ≤ε2 .. . |fn(X)−fnCP W Li (X)| ≤εn i= 1, . . . , r. (4) Then we are looking for an error bound for the trajectories such that:
kX−Xik∞≤hi(ε), i= 1, . . . , r, (5)
where hi : <n → <+ is a continuous function, ε = [ε1. . . εn]T,Xi is the approximate trajectory to X(t)running into theithsimplex andk.k∞stand for
the maximum norm.
In this way, the approximating CPWL control system will be:
˙
Xi =fCP W Li (Xi) +B·(Au(i)·Xi+B(ui)) where Xi = [X1i· · ·Xni]T is the state vector of the CPWL approximation ODE while X(t) are the trajectories of the Nonlinear ODE.2.
Next subsection develop Error Bounds in the sense of (5) which share the dynamics of the CPWL approximation ODE.
2.2 Obtaining an Error Bound for the Ap-proximate Trajectories
Taking into account (4), is possible to obtain an error bound in the spirit of (5) as follows:
|fj(X)−fjCP W Li (X)| ≤εj, j= 1, . . . , n, i= 1, . . . , r, (6)
Defining the Error for the Trajectories for theith simplex as follows:
Ei(t) =X−Xi(t). (7) Adding and subtractingB·uCP W L, uCP W L= Ai
u·Xi+Bui and considering the notation introduced in previous section:fCP W L(X) =Ai·X+Bi, then equation (6) becomes:
2This notation is providing a formal statement of equation (2)
|E˙ji−(Ai+B·Aiu)·Ei| ≤εj, j= 1, . . . , n, i= 1, . . . , r. (8)
Here we wrote Ei = [E1i· · ·Eni]T. Notice that (8) is in fact a differential inequality in Eji for each j= 1, . . . , n, however this equation can be recast in a matrix way as follows:
(
−ε≤E˙i−(Ai+B·Ai
u)·Ei≤ε,
i= 1, . . . , r, (9)
whereε= [ε1· · ·εn]T. The last step will be then to solve the Matrix Inequality in equation (9) and for that a Theorem by Coppel will be used -see [21], pp. 27-29):
Theorem 1 Lety(x)be a solution of the scalar ODE
dy(x)
dx =f(x, y)wheref(x, y)is continuous. Ifu(x)
is continuous and satisfiesu(a) ≤ y(a)and dudx(x) ≤
f(x, u)on[a, b], then:
u(x)≤y(x) (10)
Similarly, ifs(a)≤y(a)anddsdx(x) ≥f(x, s)on[a, b], thens(x)≥y(x)on[a, b].
Then the Error Bounds in equation (9) can be re-stated now: E∗1(t)≤E(t)≤E∗2(t) ˙ E∗1(t) = (Ai+B·Ai u)·Ei−ε ˙ E∗2(t) = (Ai+B·Ai u)·Ei+ε j= 1, .., n (11)
It is worth noticing that the stability issues of the Error Bound in (11) are shared by the CPWL ODE XCP W L˙ = fCP W L(XCP W L) + B · uCP W L(XCP W L). On the other hand, the point miss-ing it is to provide a way to determine when a CPWL system is stable and for that we have the following Theorem3:
Theorem 2 Given an Continuous CPWL ODE: ˙
X =Ai·X+Bifor a DomainΩwith only equilibrium
points in the interior of the simplices4with all the ma-trices Ai stable (negative definite) and such that the trajectories remains insideΩfor all x ∈ Ω, then the ODE is stable in the sense that the equilibrium points are attractive for any initial conditionx(0)∈Ω.
3The proof is given in the appendix. 4
That means to exclude the possibility for equilibrium points in the frontiers
3
CPWL AND NONLINEAR
SYS-TEMS SHARE PROPERTIES
As depicted in previous section, for systems which all the matrices Ai are stable for any simplex in the CPWL approximation ODE, then the Error Bounds in equation (11) also go to an equilibrium point showing that the Nonlinear ODE is following the CPWL ap-proxiamtion when the same CPWL controller is used in both systems.Here arises the important remark that for a suf-ficient amount of simplices (ensuring low values of ε), the properties imposed to the CPWL approximat-ing system are also Shared by the Nonlinear given system. In this way, we can design a controller to track some reference using the CPWL system, regu-late some given outputs, check stability, design opti-mal controllers, etc.
One of the most important properties in a Non-linear control system is the concept of controllability (observability), where a controlleruis required in or-der to drive the system from an initial pointx(0)to a pointx(T) for some timeT (see [23], pp. 511). In this way, is useful to introduce the concept of
quasi-controllability:
Definition 3 (ρ-Quasi-Controllability): A Nonlinear
system to the formX˙ =f(X) +g(x)·uis calledε -Quasi-Controllable if there exists a controlleru, such that the trajectories are close to the point XT in ε
whentgoes to infinity:
X(T) =XT X(0) =X0 −ρ≤limt→∞X(t)−XT ≤ρ ρ >0 (12)
Theorem 4 : Given a Nonlinear system X˙ =
f(X) +B·u and its CPWL approximation: X˙i = fi
P W L(Xi) +B·u(t), then the Nonlinear system is ρ-Quasi-Controllable.
Proof: According to the definition in (12), we only have to realize that the quantity ρ is not more than equilibrium of the Error Bound: (Ai+B ·Aiu)−1·ε obtained in equation (11), for some simplexI where
the pointX(T)belongs. ut
Is interesting to notice that the definition of Quasi-Controllability becomes the classic Controlla-bility definition when the number of simplices tend to infinity. In this way this concept is telling how possi-ble is to drive a non-linear system close to some de-sired point using a controller designed in the basis of the CPWL approximation.
4
EXAMPLE: Academic Example
A simpel example taken from [11] is the inverted pen-dulum: ( ˙ x1(t) =x2 ˙ x2(t) =−0.1·x2+sin(x1) +uCP W L (13) In this way and using 10 simplices in each co-ordinate with a range of [−1,1] in the x coordinate and[−10,10] fory to produce a CPWL of the Drift [x2, −0.1·x2+sin(x1)]0, it is possible to apply the
stabilization technique depicted in [11], pp. 102, to design a control law give byuCP W L = [−1.20 − 1.15]·[x, y]0. With this controller we , the
maxi-mum error is of0.03in absolute value, so both CPWL and Nonlinear systems are stabilized by the same con-troller.
5
Conclusion
A methodology for designing controllers for some a class of Nonlinear Control systems was presented. In this methodology an approximation of the Nonlin-ear Drift of the given Control system is approximated with a CPWL one.
Using the CPWL approximation ODE as a Paral-lel Control system, the design of a controller with any desired property (stability, optimal strategy, etc) for the Nonlinear system is then translated into the design for the CPWL one. It was shown that the same prop-erties assigned to the CPWL system are shared by the Nonlinear system when the same controller is applied to both dynamics.
An example showed an application of the theory developed, specially the idea of Quasi-controllability introduced in section 3. This concept is pointing into the direction of proving strong properties using the CPWL control system, let’s say it would be interesting to show that the controllability of CPWL and Nonlin-ear systems are equivalent.
6
Appendix
Proof of Theorem 2
Then is possible to define an auxiliary ODE as follows: Z = ( Ai·X+Bi= ˙X, X6=V ertices Z(Tvertex), Otherwise (14) then:
˙ Z =
(
Ai, X6=V ertices 0, Otherwise
whereTvertex are the instants where the trajecto-ries of the CPWL ODE is crossing from one simplex to the next one. Notice that this definition is prevent-ing problems whit the discontinuities ofZ˙ at the ver-tices. In this way, we have:
˙
Z =Ai·Z ⇔Z˙ = 1 2 ·
∂(Z0·Z)
∂x Consider next the following integral:
Z Z(t) Z(0) ˙ X0·dZ = Z t 0 ˙ X0·Z˙ ·dt= (15) = 1 2 · Z X(t) X(0) ∂(Z0·Z) ∂X 0 ·dx | {z } d(Z0·Z) In other words: Z Z(t) Z(0) ˙ X0·dZ=kZ(t)k2− kZ(0)k2
In order to solve this integral, the continuity of the variable Z(t) in the borders or frontiers in virtue of definition (14) is invoked. Then:
Z Z(t) Z(0) ˙ X0·dZ = m X i=1 Z Ti+1 Ti ˙ X0Z˙ ·dt= = 1 2 · mX−1 i=1 Z X(Ti+1) X(Ti) ∂(Z0·Z) ∂X 0 ·dx+ + Z X(t) X(Tm−1) ∂(Z0·Z) ∂X 0 ·dX.
wherek.k2 is the l2-norm (see [28] for a further
reading) andTi, i= 1, .., mare the crossing times or the times where the trajectories cross from one sim-ples to the next one.
On the other hand the integral in equation (15) also means: Z t 0 ˙ X0·Z˙ ·dt= Z t 0 ˙ X0·Ai·X˙ ·dt (16) If we requirex˙0·Ai·X <˙ 0, then this equation leads:
Z Z(t)
Z(0) x˙
0·dZ <0⇔ kZ(t)k
2− kZ(0)k2<0
Notice that only for the very special case of Piece-wise Quadratic functions asX˙0Ai·X, the requirement˙
˙
X0·Ai·X <˙ 0is equivalent to say thatAiis asymp-totically stable or negative definite (see [27]).
Focusing the aim of stability we need to consider the integral in equation (16) whitttending to infinity, which yields: ∞ X i=1 Z Ti+1 Ti ˙ X0·Ai·x˙·dt=f inite
It is well known that a necessary condition for a convergence of a an infinite sum is the limit at infinity of the general term tending to zero, in this case this leads: limi−→∞ Z Ti+1 Ti ˙ X0·Ai·x˙·dt= 0 (17) This only leaves two possibilities:
◦ X˙0·Ai·x˙ −→0ast−→ ∞
◦ limi−→∞(Ti−Ti+1) = 0
The first case is our desired objective since X˙0 ·
Ai ·x˙ → 0meansx˙ → 0, however the second pos-sibility requires a special care, it is indicating that the equilibriums are allocated in the frontiers leaving the possibility of instabilities even when the dynamics of the individuals simplices are stable as reported in [26] and [25].
u t
Acknowledgements: The research was supported
by CIC, ONICET, Universidad Nacional del Sur and Agencia Nacional de Promoci´on Cient´ıfica y T´ecnica.
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