Modeling and Control of Fuzzy, Heterogeneous and Hybrid Systems

Copyright Cl IFAC Intelligent Components and Instruments for Control Applications, Annecy, France, 1997

MODELING AND CONTROL OF FUZZY, HETEROGENEOUS AND HYBRID SYSTEMS Mikael Johansson Jorgen Malmborg Anders Rantzer Bo Bernhardsson Karl-Erik Arzen Department of Automatic Control, Lund Institute of Technology, Box 118, 8-221 00 Lund, Sweden, Email: [email protected]

Abstract: A review of some recent stability results for fuzzy, heterogeneous, and hybrid control systems. Keywords:

Nonlinear control, Fuzzy control, Lyapunov stability

1. INTRODUCTION

Increased industrial demands on quality and performance over a wide range of operating regions have led to an increased interest in nonlinear control methods during recent years. The advent of 'new' techniques such as fuzzy control, neural networks, wavelets, and hybrid systems has "amplified the interest. Nonllnear control is of interest, e.g., when the process that should be controlled is nonlinear and/or when the performance specifications are nonllnear. Basically all real processes are nonlinear, either through nonlinear dynamics or through constraints on, e.g., states and inputs. Nonlinear teclm.iques may be used for process modeling. The derived process model can be used, e.g., as the basis for model-based control design. Here, the" model may be used off-line during the design or on-line, as a part of the controller. N onlinear techniques can also be used to design the controller directly, without any process model. A variety of methods may be used to define nonlinearities, e.g, analytical equations, fuzzy systems, sigmoidal neural networks, splines, radial basis functions, wavelets, local modelsIcontrollers, discrete switching logic, lookup tables, etc. The methods can be viewed as different ways of parameterizing nonlinearities, see Figure 1. Many of the methods have been shown to be universal function approximators

for certain classes of functions. For example, fuzzy logic systems are universal approximators for continuous functions defined on compact sets [Castro, 1995]. The same holds for sigmoidal neural networks [Zbikowski et al., 1994]. Wavelets are universal approximants for the more general Besov function class [Juditsky et al., 1995]. This means that many of the

Process

Fig. 1 Different parameterizations of nonlinear con·

trollers.

methods are equivalent with respect to which nonllnearities that they can generate. Hence, it is of little value to argue whether one of the methods is better than the others if one considers only the closed loop control behaviour. From the process' point of view it is the nonlinearity that matters and not how it is parameterized. When comparing different nonlinear methods 33

there are a number of important points to consider. As pointed out, one of them is the function approximation properties of the method. Another related issue is the efficiency of the approximation method in terms of the number of parameters needed to approximate a given function . The extreme case here is the table lookup method. Of great practical importance is whether the methods are local or global. Local methods allow local adjustments. Examples of local methods are radial basis functions, splines, and fuzzy systems. How well the methods support the generation of nonlinearities from input/output data, i.e., identification/learning/training, is also of large interest. The backpropagation method is well-known in the neural network case. Other important issues are how well theoretically grounded the methods are, i.e., to what extent analysis and synthesis methods are available; how transparent the methods are, i.e., how readable the methods are and how easy it is to express prior process knowledge; the computational efficiency of the method; the availability of computer tools; and finally, subjective preferences such as how comfortable the designer/operator is with the method, and the level of training needed to use and understand the method.

the design objectives are sufficiently separated in frequency. In a similar way, many control objectives are separated in the state space, and a nonlinear controller can be patched together by several controllers optimized for separate objectives. As an example, consider the situation in Figure 2, where different controllers are used in different operating regimes. Heterogeneous

Xl

Fig. 2

controllers consist of a mixture of continuous controllers and discrete events. Thus, from a system theoretic point-of view, they may be analyzed using hybrid system theory [Branicky, 1995], [Brockett, 1993] . A heterogeneous controller can be seen as a multi-controller [Morse, 1995] that selects between a number of controllers based on some criterion checked by a supervisor. The supervisor can either select one of the controllers (hard switching) or mix the outputs of several control algorithms (soft switching). An example of a simple binary selector is a min-max selector. An example of a mixing selector could be a simple linear interpolation or a fuzzy interpolation.

Several of the methods can be written as a weighted sum of basis functions : M basis functions r-"'--.

((x)

=L

i=l

gi(X)

Different controllers in different operating regions.

Wi

'-v-' weights

The decision to switch between the controllers can have several reasons. One reason could be that the system has different dynamics in different operating regimes. The operating regime decides which controller to use. This can be compared with gainscheduling. A second reason for changing between controller could be that the task that the controller should perform changes, i.e., the performance specifications change. Another reason for changing controllers is that the dynamics of the process is timevariant. This is the motivation behind adaptive control. Related to adaptive and selftuning control is the concept of expert control or autonomous control [Astrom et al., 1986]. Here, it is the level of available process knowledge that decides which controller that should be used. By performing simple open and closed-loop auto-tuning experiments the control system builds up a series of increasingly complex process models. For each model a controller can be designed. The simpler, hopefully more robust, controllers are used as back-up controllers. Similar ideas can be used in the context of fault-tolerant distributed control systems.

This common form applies, e.g., to fuzzy systems, sigmoidal neural networks, radial basis functions, and wavelets.

2. HEXEROGENEOUSCONTROL Control design involves a number of, often conflicting, objectives. These include stability, tracking performance, load disturbance rejection, sensitivity to measurement noise, robustness to process variations, constraint satisfaction, etc. Good design methods exist for each separate issue, but most methods fall short when it comes to meeting several objectives. The idea of heterogeneous control is to combine several controllers, each designed for a separate objective, in a clever way. For linear control systems, there is a fundamental trade-off between setrpoint and load disturbance performance on one hand, and sensitivity to measurement noise on the other hand. The trade-Qff is quantified through Bode's sensitivity integral. The conflict can be resolved when 34

The heterogeneous approach also applies to process modeling. The process model may consist of a set of local models that each describes the process in a certain operating region [Johansen, 1994]. The overall model can also consist of a set of models of different granularity, each relevant for a certain control task.

3. PIECEWISE AFFINE SYSTEMS 'lb derive useful results for nonlinear systems, one must constrain the model class. Piecewise affine systems are a natural and extension of linear systems. Many control systems have piecewise affine dynamics. Examples are linear control systems subject ·to actuator saturation or dead-zones as well as logic-based switching between a set of linear controllers. Moreover, piecewise affine systems is a powerful model class for approximation of nonlinear systems.

5

5 -5

Fig. 3 Fuzzy sets and the state space partitioned into operating and interpolation regimes.

find a single globally quadratic Lyapunov function for the system. The search for a quadratic Lyapunov function can be stated as a convex optimization problem in terms of Linear Matrix Inequalities (LMls) [Boyd et al., 1994]. Efficient numerical routines for solving LMIs are publically available [Gahinet et al., 1995], and LMI based methods have been successfully applied to the design and analysis of fuzzy systems [Zhao, 1995] [Tan aka et al., 1996] .

A piecewise affine system consists of a decomposition of the state space into a set of cells, or operating regimes, Xi. Associated with each operating regime is an affine dynamics for x

E

Xi

The cells can be overlapping or non-overlapping. An actuator saturation gives disjoint operating regimes, while a hysteresis nonlinearity results in operating regimes that overlap. Another case when the operating regimes overlap is when the composite model is obtained through interpolation of locally valid models. This is the case for local model networks and the Takagi-Sugeno type offuzzy system. These systems can be written on the form

When applied to piecewise affine systems, these stability conditions are often found to be conservative in the sense that they fail to prove stability for a large class of stable systems, see [Johansson and Rantzer, 1996] for examples and [Carless, 1994] for some relevant results on quadratic stability. The conservatism of the existing approaches to quadratic stability is twofold; no information of the cell partition is taken into account and the Lyapunov function is restricted to be globally quadratic. A less conservative result has recently been suggested in [Johansson and Rantzer, 1996]. The idea is to use a piecewise quadratic Lyapunov function that is tailored to fit the cell partition of the system. The search for a piecewise quadratic Lyapunov function can also be formulated as an LMI-problem. The ideas can be extended to more general uncertain and nonlinear systems [Johansson and Rantzer, 1997], and similar techniques can also be used for performance analysis, such as L,. gain analysis of nonlinear systems [Rantzer and Johansson, 1997]. Here, the result is formulated for fuzzy systems.

M

X = Lgi(X) (AiX + ai)

(1)

i=l

The basis functions gi : x ~ [0,1] typically have the shape shown in Figure 3 (top); within operating region Xi, gi(X) = 1 and in between operating regions there is a small interpolation region where the basis functions smoothly goes to zero. This basis function shape leads to the partitioning of the state space shown in Figure 3 (bottom). The partitioning of the state space into operating and interpolation regimes applies for fuzzy systems of arbitrary dimension.

3.1 Piecewise Quadratic

~apunov

A fuzzy system (1) induces a partition of the state space into cells that are polyhedra, i.e. cells whose boundaries are given by hyperplanes

Functions

Asymptotic stability of nonlinear systems can in many cases be verified by an appropriate Lyapunov function. The standard approach for piecewise linear systems has so far been to try to

x e Xi (")Xj, i,i e I

(2)

where I is the index set of the cells. We define

35

10 to be the set of indices for cells that contain the origin, and 11 to be the set of indices for cells that do not contain the origin. Furthermore, for a given i E I we let K (i) be the index set for the dynamics used in the interpolation in cell Xj.

REMARK 1 Inequalities (3) and (5) assure positivity of the Lyapunov function

From the boundary equations (2), we construct matrices Ej E Rqx(n+l) and Ej E Rqxn such that

E

j

[~] ~ 0, Ejx

~O,

Xj,

X E

i

E

Inequalities (4) and (6) assure that the value of the Lyapunov function decreases along all system trajectories. Note that the conditions are LMIs in '1', U i , Vj , Vi and Vi.

10

We also construct matrices Fj = [F j fj] e RPx(n+l) with fj = 0 for i E 10 that satisfy for x

E

To get full freedom, we separate the Fj matrices used in the Lyapunov function parameterization from the E j matrices used in the S-procedure. In some cases, however, it is possible to let Fj = E j •

o

Xi (\ Xj i,j e I

Recall that the vector inequality w ~ 0 means that all elements of w should be nonnegative. For notational clarity, we also define

EXAMPLE 1

Consider the Takagi-Sugeno system (1), with basis functions gj(x) shown in Figure 3, and system matrices

A = [-10 I 10 where aj = 0, i E 10. Let x(t) be a continuous piecewise Cl trajectory in Vje/Xj with

L

x=

Az =

9

[-12 -2] -8

A = [-10 3 10

gk(x)(Ak X + ak),

-ll}

-10] 5

'

'

keK(i)

andgk : x

~

Although the conditions for quadratic stability fail, simulations indicate that the system is stable. Using Theorem 1, we compute a piecewise quadratic Lyapunov function that assures system stability. The level curves of this Lyapunov 0 function are indicated in Figure 4.

[0,1]. We have the following result.

THEOREM l-STABILITY OF Fuzzy SYSTEMS

If there exist a symmetric matrix T and symmetric matrices U j , Vj , V ik and Vik with nonnegative entries, such that

satisfy (3) (4)

for i Elt, k E K(i), and

o < P j - ETUjEj o > Af Pj + PiAk + ETVikEj for i E 10 , k exponentialiy.

E

Fig. 4 Simulations and level curves of the computed piecewise quadratic Lyapunov function of Example 1.

(5) (6)

Note that the above approach encompasses quadratic stability as a special case, and that the stability conditions reduces to quadratic stability when the basis functions have global support. A possible drawback of the method presented herein is that it may be required to solve

K(i), then x(t) tends to zero 0

Proof: The proof follows the lines of [Johansson and Rantzer, 1996] and is omitted here. 36

a large number of LMIs in the interpolation regions. For example, in Example 1, 5 + 7 LMIs are needed. Without interpolation only 3 + 3 LMIs are needed. The difference becomes larger if the interpolation is performed in multiple dimensions. In some cases it may therefore be of interest to exploit the alternative approach for nonlinear systems analysis given in [J ohansson and Rantzer, 1997] . The idea of that approach is to bound the system nonlinearity with piecewise affine bounds according to Figure 5~ from which simple LMI conditions for stability can be deduced. However, there is still structure to exploit and consequently there is a potential to further reduce the conservatism in stability analysis of fuzzy systems.

that, n

X = f(x, t , u*)

=L

adi(x, t)

(8)

i=1

where ai ;::: 0 satisfies L ai = 1 and where ai = o if either x ~ ni or if Vi(x,t) > minAVj(x,t)].

o

Notice that the ai's are not unique. We have the following result. THEOREM 2-STABILlTY OF HYBRID SYsTEMS

Let the system be given by (7). Introduce W as

f(x)

________

~--

The closed loop system is stable with W as a non-smooth Lyapunov function if the minswitch strategy (8) is used. 0

__----__ x

Proof" See [Malmborg et al., 1996] for proof and examples. Fig. 5

Piecewise affine bounds of a nonlinear function .

4.1 Lyapunov function transformations From a control designer's point of view the design of an hybrid control scheme using the minswitching strategy can be reduced to separate designs of n different control laws and their corresponding Lyapunov functions. Th improve performance it is often convenient to change the location of the switching surfaces. This can to some degree be achieved by different transformations of the Lyapunov functions. One example is transformations of the form

4. STABILIZING SWITCHING SCHEMES

In the previous section it was the cell partition that was used to determine the piecewise quadratic Lyapunov function and hence prove stability. An alternative approach is to let the stability requirements determine when to switch between different controllers. It is wellknown that switching between stabilizing controllers may lead to .a n unstable closed loop system. Consider the system

x = Ui

=

f(X,t,Ui)

(9)

(7)

where gi(") are monotonously increasing functions.

Ci(X,t)

where the Ci(X) represent different controllers. In a hybrid control system we would like to switch to different controllers in different regions of the state space or in different operating modes. There exist some switching schemes that guarantee stability. One of these is the minswitch strategy described in [Malmborg et al., 1996]. Here, a number of stabilizing controllers, Ci, are designed for the system 7. For each controller Ci, an operating region, Oi, is defined and a Lyapunov function, Vi, is derived. At every moment the supervisor selects the controller with the smallest value of its Lyapunov function. The controllers can be of different types and they need not share the same state space.

4.2 Real-time implementation A control system based on the min-switching strategy has been successfully implemented, see [Malmborg and Eker, 1997]. The system consists of two water tanks in series. The goal is to control the level of the second tank. Two controllers are used, a time~ptimal controller to achieve fast set point responses and a PID controller for the regulation problem. Each time a new reference signal command is given, the time-optimal switching curve corresponding to a linearized model of the tank around the new reference signal is calculated. The time-optimal controller is used until the process is sufficiently close to the new setpoint, when the system switches to PID control.

DEFINITION l-MIN-SwlTcmNG STRATEGY Let fi(X , t) be the right-hand side of (7) when control law Ci is used. Use a control signal u· so

37

5. CONCLUSIONS Increasing complexity and performance requirements motivate the heterogeneous approach to control system design. There are many interesting open problems in heterogeneous control theory. One problem is the design of stable switching schemes, i.e., how one should combine several controllers so that the multi-controller system is stable and meets the design objectives. Another problem is to determine in what cases the controllers can be optimized individually, and in what cases it is worthwhile to do a combined controller optimization.

Gahinet, P., A. Nemirovski, A. J. Laub, and M. Chilali (1995): LM1 CoIltrol lbolboz for use with MatIab. The Mathworks Inc. Johansen, T. A. (1994): Operating Regime Based Process ModeIing 8.1ld Iden.tificatiOIl. PhD thesis ITK-rapport 1994:109-W, NOIWegian Institute of Technology, Trondheim. Johansson, M. and A. Rantzer (1996): "Computation of piecewise quadratic Lyapqnov functions for hybrid systems." Technical Report !SRN LUTFD2fI'FRT--7459--SE, Department of Automatic Control. Also available at http://vvv.contro1.1th.se/-rantzer. Johansson, M. and A. Rantzer (1997): "Computation of piecewise quadratic Lyapunov functions for hybrid systems." In Europes.n. COIltrol Comeren.ce, ECC97. (in submission).

Heterogeneous control has strong relationships to function approximation based nonlinear modeling and control. A number of such methods exist, e.g., neural networks, fuzzy systems, and wavelets. Several of the methods have strong similarities with respect to which nonlinearities that can be generated. It is the authors' belief that automatic control research would benefit from a unification of the subtly different approaches to approximation based nonlinear control instead of allowing a continuous recasting of well-known results from one field into the other.

Juditsky, A., H. Hjalmarsson, A. Benveniste, B. Delyon, L. Ljung, J. Sjoberg, and Q. Zhang (1995): "Nonlinear black-box models in system identification: Mathematical foundations." Automatica, 31:12, pp. 1725-1750. Malmborg, J., B. Bernhardsson, and K J. Astrom (1996): "A stablizing switching scheme for multi· controller systems." In Proceedings of tile 1996 '!hennaJ IFAC World Congress, IFAC'96, vol· ume F, pp. 229-234, San Francisco, California, USA. Elsevier Science.

Acknowledgements This work has been supported by the Esprit project FAMIMO, the Institute of Applied Mathematics, Sweden, and by NUTEK, the Swedish National Board for Industrial and Technical Development, under contract 95-02540.

Malmborg, J. and J. Eker (1997): "Hybrid control of a double tank system." In CCA'97, Hartford, CT. (in submission). Morse, A. S. (1995): "Control using logic-based switching." In Isidori, Ed., Tren.ds in Control. A European Perspective, pp. 69-113. Springer. Rantzer, A. and M. Johansson (1997): "Piecewise linear quadratic control." In AmeriC8.1l Control CoIlferen.ce, ACC'97. 'Ib appear.

REFERENCES Astrom, K J., J. J. Anton, and K-E. ken (1986): "Expert control" Automs.tica, 22:3, pp. 277-286.

Tanaka, K, T. Ikeda, and H. O... Wang (1996): "Robust stabilization ofa class of.uncertain nonlinear systems via fuzzy control: Quadratic stabilizability, Hoo control theory and linear matrix inequalities." IEEE 'I'ransactioIlS OIl Fuzzy Systems, 4:1, pp. 1-13.

Boyd, S., L. El Ghaoui, E. Feron, and V. Balakrishnan (1994): Linear Ms.triz JnequaJities in System 8.1ld CoIltrol Theory. Siam Studies in Applied Mathematics.

Zbikowski, R, K Hunt, A. DzieliDski, R MurraySmith, and P. Gawthrop (1994): CA Review of Advances in Neural Adaptive Control Systems." Technical Report, Daimler-Benz AG and University of Glasgow.

Branicky, M. S. (1995): Studies in Hybrid Systems: Modeling, .AIJ.s.Jysis 8.1ld CoIltroL PhD thesis, MIT.

Brockett, R W. (1993): "Hybrid models for motion control systems.- In Trente1man and Willems, Eds., Essays OIl CoIltrol: Perspectives in the Theory 8.1ld its Applicati01l. Birkhii.user.

Zhao, J. (1995): System ModeJing, Identification 8.1ld CoIltrol using Fuzzy Logic. PhD thesis, UeL, Universite Catholique de Louvain.

Castro, J. (1995): "'Fuzzy logic controllers are universal approximators.- IEEE TnmsS.ctiOIlS on Systems, Man and Cybemetics, April, pp. 62~5. Corless, M. (1994): "'Robust stability analysis and controller design with quadratic lqapunov functions.· In Zinober, Ed., Variable Structure 8.1ld Lyapunov Control, Lecture notes in Control and Information Sciences, chapter 9, pp. 181-203. Springer Verlag. 38