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Abstract
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Computer Science
Modeling and Simulation

Neural nets as systems models and controllers

Profile image of Eduardo D. SontagEduardo D. Sontag

1992

Abstract

Repr. from Proc. Seventh Yale Workshop on Adaptive and Learning Systems, pp. 73-79, May 1992. This paper briefly surveys some recent results relevant to the suitability of "neural nets" as models for dynamical systems as well as controllers for nonlinear plants. In particular, it touches upon questions of approximation, identifiability, construction of feedback laws, classification and interpolation, and computational capabilities of nets. No discussion is included of "learning" algorithms, concentrating instead on representational issues.

FAQs

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What explains the role of neural nets in modeling dynamic systems?add

The research shows that neural nets can approximate a wide class of nonlinear systems, demonstrating universal identification capabilities for various input-output configurations. For instance, recurrent nets are capable of simulating control systems accurately across compact subsets of their state space.

How do neural nets compare to traditional controllers in practical applications?add

The findings suggest that neural nets can handle more complex control tasks, such as gain scheduling, with improved performance. Specifically, using dynamic feedback and piecewise linear regulation yields better adaptability in systems like reconfigurable robotic manipulators.

When are two hidden layers necessary in neural networks for inverse problems?add

The paper establishes that two hidden layers are sufficient to solve general inverse problems involving continuous functions. One hidden layer may not suffice due to the limitations in approximating characteristic functions of bounded polytopes.

What defines the universality of activation functions in neural networks?add

An activation function is considered universal if it can approximate every continuous function on compact intervals. Specifically, functions that are continuous and nonconstant, such as sigmoid functions, display this universality property, allowing for comprehensive function approximation.

What methodological innovations improve neural network performance in control theory?add

The study emphasizes using recurrent nets with targeted architectures that integrate nonlinearity, which enhances their ability to function as universal models. Enhanced feedback mechanisms through dynamic systems allow neural nets to maintain stability and control in complex environments.

Figures (5)
Figure 2: Static Net Computing the Function In the static case, we consider nets formed by inter- connections without loops, as otherwise the behavior is not always well-defined; we call these “feedforward” nets, in contrast to the terms “feedback,” “recurrent,” or “dynamic” nets used in the general case.
Figure 2: Static Net Computing the Function In the static case, we consider nets formed by inter- connections without loops, as otherwise the behavior is not always well-defined; we call these “feedforward” nets, in contrast to the terms “feedback,” “recurrent,” or “dynamic” nets used in the general case.
Figure 3: Dynamic Net Representing the System 20(30(5u1 —u2)+20 (ui +2u2+1)+1)+50(—30(u1+2u2+1)—1)
Figure 3: Dynamic Net Representing the System 20(30(5u1 —u2)+20 (ui +2u2+1)+1)+50(—30(u1+2u2+1)—1)
of the state space or output-value space the current measurement is in; see Figure 4. This is known some- times as “gain scheduling” or, if switching betweer controllers is replaced by a soft switching or blending of control laws, “fuzzy control.” It is a useful contro! architecture when dealing with plants exhibiting qual- itatively different modes of operation, such as reconfig- urable aircraft or robotic manipulators under variable loads. In the first role, nets typically compute func- tions with continuous values, while in the second case nets are used as classifiers, producing a binary or more generally a finite-valued output. Mathematically, the first case gives rise to the study of questions of function approximation and interpolation, while the second fo- cuses on Classification. We will discuss results relating with both of these aspects. One of the conclusions is that static nets are capable of controlling fairly general nonlinear plants, if state feedback is available.
of the state space or output-value space the current measurement is in; see Figure 4. This is known some- times as “gain scheduling” or, if switching betweer controllers is replaced by a soft switching or blending of control laws, “fuzzy control.” It is a useful contro! architecture when dealing with plants exhibiting qual- itatively different modes of operation, such as reconfig- urable aircraft or robotic manipulators under variable loads. In the first role, nets typically compute func- tions with continuous values, while in the second case nets are used as classifiers, producing a binary or more generally a finite-valued output. Mathematically, the first case gives rise to the study of questions of function approximation and interpolation, while the second fo- cuses on Classification. We will discuss results relating with both of these aspects. One of the conclusions is that static nets are capable of controlling fairly general nonlinear plants, if state feedback is available.
A constant vector c could be added to the right- hand side of equation (2), to represent what are usu- ally called “biases” in the neural net literature, that is, in order to allow affine rather than just linear com- binations of states and inputs, leading to equations Az = o,(At+ Bu-+c). It is often simpler to ig- nore this term by first adding a constant coordinate xo(t) = pw, where u # 0 and then replacing each en- try c; by the linear term (c¢;/{1)%, which can then be absorbed into the A matrix. (In continuous-time, as- suming that o(0) = 0, then 29 evolves according to &o = 0(0); in discrete-time, we may take any nonzero element in the image of o, = a(v), and then use the update #7 = o((v/p)20).) where A € IR”*”, B € R”*™, and C € R’*”, and A =x* or = & in discrete or continuous time respec- tively. We call these o-systems or recurrent nets.
A constant vector c could be added to the right- hand side of equation (2), to represent what are usu- ally called “biases” in the neural net literature, that is, in order to allow affine rather than just linear com- binations of states and inputs, leading to equations Az = o,(At+ Bu-+c). It is often simpler to ig- nore this term by first adding a constant coordinate xo(t) = pw, where u # 0 and then replacing each en- try c; by the linear term (c¢;/{1)%, which can then be absorbed into the A matrix. (In continuous-time, as- suming that o(0) = 0, then 29 evolves according to &o = 0(0); in discrete-time, we may take any nonzero element in the image of o, = a(v), and then use the update #7 = o((v/p)20).) where A € IR”*”, B € R”*™, and C € R’*”, and A =x* or = & in discrete or continuous time respec- tively. We call these o-systems or recurrent nets.

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References (13)

  1. Albertini, F., and E.D. Sontag, "For neural networks, function determines form," submitted.
  2. Cybenko, G., "Approximation by superpositions of a sigmoidal function," Math. Control, Signals, and Sys- tems 2(1989): 303-314.
  3. Hornik, K., "Approximation capabilities of multilayer feedforward networks," Neural Networks 4(1991): 251- 257.
  4. Maass, W., G. Schnitger, and E.D. Sontag, "On the computational power of sigmoid versus boolean thresh- old circuits," Proc. of the 32nd Annual Symp. on Foun- dations of Computer Science, 1991: 767-776.
  5. Matthews, M., "On the uniform approximation of non- linear discrete-time fading-memory systems using neu- ral network models," Ph.D. Thesis, E.T.H. Zurich, Diss. ETH No. 9635, 1992.
  6. Polycarpou, M.M., and P.A. Ioannou, "Identification and control of nonlinear systems using neural network models: Design and stability analysis," Report 91-09-01, Sept. 1991, Dept. of EE/Systems, USC, Los Angeles.
  7. Romanik, K., "Approximate testing and learnability," in Proc. Fifth ACM Workshop on Computational Learn- ing Theory, Pittsburgh, July 1992.
  8. Siegelmann, H.T., and E.D. Sontag, "On the computa- tional power of neural nets," in Proc. Fifth ACM Work- shop on Computational Learning Theory, Pittsburgh, July 1992; see also SYCON Report 91-11, Rutgers Cen- ter for Systems and Control, November 1991.
  9. Siegelmann, H.T., and E.D. Sontag, "Analog compu- tation, neural networks, and circuits," submitted.
  10. Sontag, E.D., "Nonlinear regulation: The piecewise linear approach," IEEE Trans. Autom. Control AC- 26(1981): 346-358.
  11. Sontag, E.D., "Feedforward nets for interpolation and classification," J. Comp. Syst. Sci., to appear, 1992.
  12. Sontag, E.D., "Feedback Stabilization Using Two- Hidden-Layer Nets," in Proc. Amer. Automatic Control Conference, Boston, June 1991, pp. 815-820. To appear in IEEE Trans. Neural Networks.
  13. Sontag, E.D., Mathematical Control Theory: Deter- ministic Finite Dimensional Systems, Springer, New York, 1990.

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