Direct Adaptive Control using Single Network Adaptive Critic Swagat Kumar∗ , Radhakant Padhi† and Laxmidhar Behera∗ ∗

Department of Electrical Engineering Indian Institute of Technology Kanpur, Uttar Pradesh, India - 208 016 † Department of aerospace engineering Indian Institute of Science, Bangalore, India - 560 012 Email: [email protected], [email protected], [email protected]

Abstract—An optimal control law for a general nonlinear system can be obtained by solving Hamilton-Jacobi-Bellman equation. However, it is difficult to obtain an analytical solution of this equation. In this paper, we propose a direct adaptive control method for affine systems where the optimal value function is approximated using a critic network and the weight update law is derived so as to satisfy the HJB equation. This provides an alternative method to solve the HJB equation in real-time and thus paves the way for introducing some notion of optimality to direct adaptive control paradigm. The efficacy of the scheme is demonstrated for a linearized mass-spring-damper system and its performance is compared with that of LQR.

I. I NTRODUCTION In case of nonlinear systems, the main focus of any control design process is to ensure stability of the system while achieving good trajectory tracking. Many a times, simple stability of the system is not just good enough. The cost of implementing the control may render it impracticable. This gives rise to optimal control methodologies where one tries to design controller which minimizes certain performance indices. While optimal control theories are quite well-established, its application to control of nonlinear systems has been limited owing to the mathematical complexity involved in finding a closed form solution. Bellman’s dynamic programming [1], [2] treats such optimal control problems as a multistage decision making process where a decision is chosen from a finite number of decisions based on some optimization criterion. However, dynamic programming is an off-line approach where one starts at the end point and moves in a backward direction. Adaptive critic solves this dynamic programming problem in forward direction and hence can be implemented on-line. Adaptive critic based methods have two components - an actor which computes the control action and a critic which evaluates its performance. Based on the feedback received from the critic, the actor improves its performance in the next step. Various architectures as well as learning algorithms for actor and critic have been proposed in last few years. An interested reader may refer to [3] and [4] for details. Adaptive critic based methods have been used to solve various realworld problems like generators [5], [6], distributed parameter systems [7], aircraft control [8], [9], [10], automobiles [11] etc. Quite recently, Padhi et. al. [12] introduced a simplified version of adaptive critic architecture which uses only one

network instead of two required in a standard adaptive critic design. This architecture is called “single network adaptive critic (SNAC)”. This architecture can be applied to a class of systems where control input can be expressed explicitly in terms of state and costate variables. In another approach, nonlinear feedback controller is synthesized by appropriate feed-forward neural network to satisfy the corresponding nonlinear HJB equation in a restricted domain of the state space [13], [14], [15]. In these methods, the networks are trained off-line using either back propagation [13] or a variant of back-propagation through time [14] to learn the optimal cost function. Unlike earlier approaches where a discrete-time approach is taken for training, Cheng et.al. [15] use a continuous time update for weights. However, the network training is carried in an off-line manner. In neural network based direct adaptive control, the weights of a neural network are updated so as to ensure stability of the closed loop system while tracking a desired trajectory. In other words, NN based direct adaptive control schemes do not involve any a priori system identification process. Interested readers may refer to [16], [17] to know more about NN based direct adaptive control schemes. This makes them preferable in case of real-time applications where online training is needed. They are also advantageous owing to the fact that they are less susceptible to modeling inaccuracies. NN based direct adaptive control schemes have been successfully applied to a large class of nonlinear control affine systems. However, their applicability get restricted owing to the requirement that the nonlinear function associated with control, g(x) (refer to equation (1)) must be known beside being non-zero for all values of x. That is, the uncertainty in the knowledge of g(x) makes the problem very difficult [18]. Before talking about the approach presented in this paper, it would be worthwhile to summarize the above discussion as follows: • Almost all critic based control schemes involve an offline training process to solve the optimal control problem and hence can not be implemented in real-time. • Most of the present critic based schemes are implemented in discrete-time and hence need a rigorous convergence analysis. Stability analysis during training process also gets complicated owing to the discrete nature of problem.

Neural network based direct adaptive control schemes can be implemented online and do not involve cumbersome off-line training processes. However they do not include any notion of optimality. • NN based direct adaptive control allow for uncertainty in the knowledge of system dynamics. Its application to affine control problems get restricted owing to the requirement of a known and non-zero nonlinear function g(x). In this paper, we introduce a direct adaptive control scheme that accounts for optimality as well. It is a well known fact that any function which satisfies the Hamilton-Jacobi-Bellman equation is optimal. However, finding an analytical solution of HJB equation is usually very difficult even for a moderately sized system. We approximate this optimal cost function using a critic network which may be a polynomial function or a neural network and then derive continuous-time weight update law so as to satisfy the HJB equation. This gives rise to an under-determined linear least square problem which can be solved accurately using standard numerical routines. Following a similar analysis as in Goh’s paper [13], we show that the system is stable in the sense of Lyapunov and in the process various convergence conditions are discussed. Some of the advantages of this approach may be enumerated as follows: • It does not involve any off-line training process and hence can be implemented online. Theoretically, it is one-shot approach, where the optimal control is achieved instantaneously. Various issues related to this are discussed later in the paper. • This control scheme removes the restriction on g(x) present in current direct adaptive schemes owing to the fact that the control law does not include the term g −1 . Hence, this scheme can be applied to a wider class of problems. The paper is organized as follows. The proposed scheme is presented in the next section followed by its stability analysis in section III. The simulation results are provided in section IV followed by conclusion in section V. •

II. D IRECT

ADAPTIVE CONTROL USING

SNAC

Consider a nonlinear control-affine system given by x˙ = f (x) + g(x)u

(1)

The task is to find a control that minimizes following cost function with zero terminal condition: Z tf J= ψ[x(τ ), u(τ )]dτ (2) 0

where the utility function ψ is given by ψ(x, u) =

1 T [x Qx + uT Ru] 2

(3)

Consider a Hamiltonian given by H(x, λ∗ , u) = ψ(x, u) + λ∗T [f (x) + g(x)u]

(4)

where λ∗ = ∂J ∂x . The optimal control is obtained from the necessary condition given by ∗

∂ψ ∂ ∂H = + λ∗T [f (x) + g(x)u] = 0 (5) ∂u ∂u ∂u This gives following optimal control equation for affine system (1): u = −R−1 gT λ∗ (6) Note that the control law does not include a term g −1 which is a common to most direct adaptive control schemes. Substituting the value of u into (4), we get H(x∗ , λ∗ , u∗ ) =

1 1 T x Qx + λ∗T gR−1 gT λ 2 2 + λ∗T [f − gR−1 gT λ∗ ] (7)

On simplification, we have following optimal Hamiltonian : 1 T 1 x Qx − λ∗T gR−1 gT λ∗ + λ∗T f 2 2 1 T 1 = x Qx − λ∗T Gλ + λ∗T f (8) 2 2 We know that the optimal cost function J ∗ (x, t) must satisfy the Hamilton-Jacobi-Bellman (HJB) equation given by H∗

=

∂J ∗ ∂J ∗ + min H(x, , u, t) = 0 (9) u ∂t ∂x It provides the solution to the optimal control problem for general nonlinear dynamical systems. However, the analytical solution to the HJB equation is difficult to obtain in most cases. The HJB equation is a necessary condition of optimality in the sense that the minimum cost function must satisfy this equation. Moreover, it is a sufficient condition in the sense that if there exists a cost function J s (x, t) which satisfies the HJB equation, then J s (x, t) is the minimum cost function, i.e., J s (x, t) = J ∗ (x, t) [1, ch. 6, pp. 286-287]. Combining (8) and (9) we can say that, in case of affine systems (1), the optimal cost function must satisfy following nonlinear dynamic equation :  T  ∗ T 1 ∂J ∗ ∂J ∗ ∂J ∂J ∗ 1 T + x Qx− G + f = 0 (10) ∂t 2 2 ∂x ∂x ∂x Since, the analytical solution of the above equation is difficult, we take a different approach where we approximate the optimal cost function using a neural network as shown below: V (x, t) = y 2 = [h(w, x)]2

(11)

where y = h(w, x) is a neural network with input vector x, weight vector w and a scalar output y. The neural network is selected such that the following condition is satisfied. V (0, t) = 0 (12) ∂V (0, t) = 0 (13) ∂x This necessitates proper selection of network architecture and activation function for the neural network such that we have h(w, 0) = 0. The need of this condition would become clear

in a short while. For this cost function to be optimal, satisfy the HJB equation (10). This gives  T ∂V ∂V + ψ(x, u) + [f + gu] ∂t ∂x  T T  ∂V ∂V ∂V 1 T 1 ∂V G f + x Qx − + ∂t 2 2 ∂x ∂x ∂x  T T  1 1 ∂V ∂V ∂V ∂V ˙ + xT Qx − w + G f ∂w 2 2 ∂x ∂x ∂x

it must

= 0 = 0 = 0 (14)

This gives following weight update law:  T T  ∂V ∂V 1 1 ∂V ∂V ˙ = − xT Qx + G f (15) − w ∂w 2 2 ∂x ∂x ∂x  T  T ∂h ∂h 1 T ∂h ∂h 2 ˙ = − x Qx + 2h 2h G f w − 2h ∂w 2 ∂x ∂x ∂x (16) ˙ so that the above scalar equation is The task is to find w satisfied. This is an under determined problem with number of equations less than the number of variables to be estimated. ˙ which Though, there are infinitely many solutions for w would exactly satisfy the above equation, we seek the one ˙ 2 . The problem is referred to as finding which minimizes kwk minimum norm solution to an under determined system of linear equations. Lapack’s [19] linear least square solver is used to solve this equation. The control scheme is shown in figure 1. Since this control scheme can be implemented in real time, we call it ‘real-time adaptive critic (RTAC)’. The blocks are self-explanatory. u Optimal Control

Plant Dynamics

x

time-dependence. Differentiating it with respect to time and using the optimal cost dynamics (14), we get T  ∂V ∂V ˙ + V = x˙ ∂t ∂x  T ∂V ∂V + [f + gu] = ∂t ∂x 1 1 = −ψ(x, u) = − xT Qx − λT Gλ (17) 2 2 where λ = ∂V ∂x . Lyapunov stability theorem states that if V (x, t) is a locally positive definite function with a negative definite time derivative V˙ (x, t), then the equilibrium point x = 0 is stable. Asymptotic stability analysis of non-autonomous system is generally difficult than that of autonomous systems, since it is usually difficult to find positive definite Lyapunov functions with negative definite derivative. This difficulty can be partially remedied by using Barbalat’s lemma. The stability analysis using Barbalat’s lemma can be done using following ’Lyapunov-like lemma’ [20, ch. 4] Lemma 1. If a scalar function V (x, t) satisfies the following conditions • V (x, t) is lower bounded • V˙ (x, t) is negative semi-definite • V˙ (x, t) is uniformly continuous in time then V˙ (x, t) → 0 as t → ∞. In our case, the Lyapunov function candidate (11) is lowerbounded and V˙ is negative definite because V˙ < 0 ∀x 6= 0 and V˙ = 0 when x = 0. This is due to the fact that ∂h ∂V = 2h (18) λ = ∂x ∂x and, λ = 0 when x = 0 ∵ h(w, 0) = 0 Moreover, V˙ is bounded. Hence by above lemma we have, V˙ → 0 and thus (x, λ) → 0 as t → ∞.

λ

V ∂ ∂x

Discussion:

Critic

˙ w HJB weight update Fig. 1.

Real time adaptive critic scheme

III. S TABILITY A NALYSIS In order to analyze the stability of the network, we consider (11) as a Lyapunov function candidate with initial conditions given by (12)-(13). Because of time-varying weight parameters, we have a non-autonomous system and thus the Lyapunov function candidate is considered to have explicit

It is necessary at this stage to discuss some issues related to the weight update law given by (16). Following points may be noted about this control scheme: ˙ as long as • The equation (16) gives a finite value for w ∂h k 6= 0. The Lyapunov analysis tells us that with kh ∂w this weight update law, the state x → 0 as t → ∞. When x = 0, the right hand side of equation (16) becomes ˙ = 0 and hence weights attain a finite zero leading to w value. This ensures the boundedness of network weights. However, it is necessary to find the conditions for which ∂h kh ∂w k 6= 0. • This scheme does not involve any off-line training process for the neural network. • This scheme paves the way for computing the solution of HJB in real-time. • The performance of algorithm depends on proper initialization of weights. In case of open loop unstable systems,

the initial weights must be selected so that the initial controller stabilizes the system. 0.9

IV. S IMULATION We demonstrate the scheme for a linear mass-spring-damper system given by        0 x1 0 1 x˙ 1 u (19) + = 1 x˙ 2 −0.4 −0.1 x2

0.3

States

This is an open loop stable system and the task is to find a control that minimizes the cost function (2) with following parameters:   1 0 Q= R=1 0 1

0 -0.3

The steady state Linear Quadratic Regulator (LQR) control for this problem is given by u = −0.67703x1 − 1.43755x2

x1, RTAC x2, RTAC x1, LQR x2, LQR

0.6

-0.6

(20)

-0.9 0

Two cases are considered - in one, the optimal cost function is approximated using a linear network while in other, it is approximated with a feed-forward network.

2

4

8

6

10

Time (sec) Fig. 2.

Case I: State trajectories

Case I: Linear Critic Network 0

In this case, we approximate the optimal cost function as The Jacobin terms are given as follows:   ∂V x = 2(w1 x1 + w2 x2 ) 1 x2 ∂w   ∂V w = 2(w1 x1 + w2 x2 ) 1 w2 ∂x

(21)

RTAC LQR

-2

Control Input

V (x, t) = h(w, x)2 = (w1 x1 + w2 x2 )2

(22) (23)

The critic satisfies the initial conditions (12)-(13). The weight update law is derived as in (16). The simulation results are shown in Figures 2-5. The comparison is shown with respect to LQR control. Figure 2 shows the evolution of state trajectories and the control inputs are shown in Figure 3. The evolution of weights w1 and w2 is shown in Figure 4. As it can be seen, the weights settle down to a constant value.

-4 -6 -8 -10 -12 0

1

2

3

4

5

Time (sec)

Case II: MLP as a critic network Fig. 3.

Case I: Control input comparison

In this case we consider 2-10-1 feed-forward network with a ’tansig’ activation function for hidden neurons and ’linear activation’ for output layer. The neural network may be represented as follows: h(w, x) = W2 φ(W1 x)

(24)

where W1 and W2 represent weight matrices between input and hidden layer and hidden layer and output network respectively and φ(.) = [φ1 , . . . , φi , . . . , φn ]T is the hidden layer vector and φi (x) = tanh(x). The optimal cost approximation and its Jacobian matrices are given by V ∂V ∂w ∂V ∂x

= h2 (w, x) T  ∂h ∂h = 2h | ∂w1 ∂w2

(25)

= 2hW2 Φ0 W1

(27)

(26)

where

φ01 0  Φ0 (.) =  .  .. 

0

0 φ02

... ... .. .

0 0 .. .

0

. . . φ0n

    

and w1 and w2 are vector representations of weight matrices W1 and W2 and w = [w1 w2 ]T . It is to be noted that the above critic structure (25)-(27) satisfy the initial conditions (12)-(13). Hence, the weight update law may be derived as shown in (16). The simulation results are shown in Figures 6 - 9.

4.8 1 x1, RTAC x1, LQR x2, RTAC x2, LQR

0.5

4.4

States

Weights

4.6

4.2

0

-0.5

4

W1 W2

3.8 0

-1

0.1

0.2

0.3

0.4

0.5

Time (sec) -1.5 0 Fig. 4.

5

10

20

15

Time (sec)

Case I: Weights of the critic network

Fig. 6.

Case II: States trajectories comparison

3 RTAC LQR

0.5

2

0

1.5

-0.5

Control Input

Optimal Cost

2.5

1 0.5 0 0

-1 -1.5 -2

2

4

6

8

10

Time (sec) Fig. 5.

RTAC LQR

-2.5 0

5

10

15

20

Time (sec)

Case I: Comparison with LQR : Optimal Cost function Fig. 7.

Case II: Control input comparison

Discussion: •



It is seen that the state and control trajectories do not closely match with those of LQR control. It is because one needs to ensure that various conditions discussed in III are met while using this scheme. Existence of various local minima can not be ruled out at this stage. Depending on initial values of weight parameters, it is possible to get a closer approximation. Moreover, in direct adaptive control, the emphasis is on stability than optimality. A deeper analysis would be presented in future versions of this paper. The performance is even worse in Case II where an MLP is being used for approximating the optimal value func-

tion. First of all, a nonlinear controller is not needed for a linear network and hence a good performance can not be expected. Secondly, the number of local minima is higher in case of MLP than in linear controller. Performance of neural network also depends on choice of suitable activation functions which need to be studied in detail. Use of MLP would be more effective in case of nonlinear systems which necessitate a nonlinear controller. V. C ONCLUSION In this paper, a direct adaptive control scheme based on solution of HJB equation is proposed. The optimal cost function

1.2

for her valuable suggestions and feedback. RTAC LQR

R EFERENCES

Optimal Cost

0.9

0.6

0.3

0 0

5

10

15

20

Time (sec) Fig. 8.

Case II: Optimal Cost function

3.8

|| W ||

3.78

3.76

3.74

3.72 0

5

10

15

20

Time (sec) Fig. 9.

Case II: Evolution of weights

is approximated using a neural network and the weight update law is derived so as to satisfy the time-varying HJB equation. Through time-varying Lyapunov analysis, it is shown that the equilibrium point x = 0 is asymptotically stable. Even though, the scheme is applicable to general class of nonlinear control affine systems, simulation results are provided for a linear mass-spring-damper system only as a demonstration exercise and nonlinear problems would be dealt with in future versions of this paper. This scheme can be implemented in real-time and introduces optimality into direct adaptive control paradigm. VI. ACKNOWLEDGMENT Authors would like to thank Indrani Kar, a senior PhD student at Department of Electrical Engineering, IIT Kanpur

[1] D. S. Naidu. Optimal Control Systems. CRC Press, 2003. Chapter 5, Discrete-time optimal control systems. [2] A. E. Bryson and Y. C. Ho. Applied Optimal Control. Taylor and Francis, 1975. [3] D. V. Prokhorov and D. C. Wunsch II. Adaptive critic designs. IEEE Transactions on Neural Networks, 8(5):997–1007, September 1997. [4] J. Si, A. G. Barto, W. B. Powell, and D. Wunsch II, editors. Handbook of learning and Approximate Dynamic Programming, chapter 3. IEEE Press, 2005. [5] P. Jung-Wook, R. G. Harle, and G. K. Venayagamoorthy. Adaptivecritc based optimal neurocontrol for synchronous generators in a power system using mlp/rbf neural networks. IEEE Transactions on Industry Applications, 39(5):1529–1540, October 2003. [6] G. K. Venayagamoorthy, R. G. Harley, and D. C. Wunsch. Implementation of adaptive critic-based neurocontrollers for turbogenerators in a multimachine power system. IEEE Transactions on Neural Networks, 14(5):1047–1064, September 2003. [7] R. Padhi, S. N. Balakrishnan, and T. Randolph. Adaptive critic based optimal neuro control synthesis for distributed parameter systems. Automatica, 37:1223–1234, 2001. [8] S. N. Balakrishnan and V. Biega. Adaptive critic based neural network for aircraft optimal control. Journal of Guidance, Control and Dynamics, 19(4):893–898, July-August 1996. [9] D. Han and S. N. Balakrishnan. Adaptive critics based neural networks for agile missile control. Journal of Guidance, Control and Dynamics, 25:404–407, 2002. [10] J. C. Neidhoefer and K. Krishnakumar. Intelligent control of nearautonomous aircraft missions. IEEE Transactions on Systems, Man and Cybernetics - Part A: Systems and Humans, 31(1):14–28, January 2001. [11] N. V. Kulkarni and K. Krishna Kumar. Intelligent engine control using an adaptive critic. IEEE Transactions on Control System Technology, 11(2):164–173, March 2003. [12] R. Padhi, N. Unnikrishnan, X. Wang, and S. N. Balakrishnan. A single network adaptive critic (SNAC) architecture for optimal control synthesis for a class of nonlinear systems. Neural Networks, Science Direct, Elsevier, 19:1648–1660, 2006. [13] C. J. Goh. On the nonlinear optimal regulator problem. Automatica, 29(3):751–756, 1993. [14] H. Bersini and V. Gorrini. A simplification of the backpropagationthrough-time algorithm for optimal neurocontrol. IEEE Transactions on Neural Networks, 8(2):437–441, March 1997. [15] T. Cheng, F. L. Lewis, and M. Abu-Khalaf. A neural network solution for fixed-final time optimal control of nonlinear systems. Automatica, 43(3):482–490, March 2007. [16] C. Kwan, F. L. Lewis, and M. Dawson. Robust neural-network control of rigid-link electrically driven robots. IEEE Transactions on Neural Networks, 9(4):581–588, July 1998. [17] C. Kwan and F. L. Lewis. Robust backstepping control of induction motors using neural networks. IEEE Transactions on Systems, Man and Cybernetics, Part A, 30(6):753–766, November 2000. [18] Indrani Kar and Laxmidhar Behera. Neural network based direct adaptive control for a class of affine systems. In IEEE International symposium on Intelligent Control, Munich, Germany, 2006. [19] Lapack, linear algebra package. http://www.netlib.org/lapack. [20] J. J. E. Slotine and W. Li. Applied Nonlinear Control. Prentice Hall, New Jersey, 1991.

Direct Adaptive Control using Single Network Adaptive ...

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