Stable state dependent Riccati equation neural ...

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Int. J. Modelling, Identification and Control, Vol. 28, No. 3, 2017

Stable state dependent Riccati equation neural observer for a class of nonlinear systems Amin Sharafian and Reza Ghasemi* Department of Electrical Engineering, University of Qom, Qom, Iran Email: [email protected] Email: [email protected] *Corresponding author Abstract: This paper proposes a new methodology for state estimation of a class of nonlinear systems. Owing to hard nonlinearity of some complex systems, the performance of linearisation methods is limited, thus this paper focuses on a new technique for nonlinear state estimation based on the combination of the state dependent Riccati equation (SDRE) and the neural network. SDRE technique is adapted to bring the certain nonlinear parts of system into linear-like structure and the unknown nonlinearities are estimated by artificial neural network whose weights are adjusted with guaranteed stability of the closed-loop system. This technique is strongly distinguishing the effect of uncertain nonlinearities and unmodelled dynamics to prevent divergence of state estimation error. Both the stability of the closed-loop system and uniform ultimate boundedness of the observer error are guaranteed based on Lyapunov theory. Keywords: state dependent Riccati equation; SDRE; state estimation; neural observer; nonlinear systems; Riccati equation. Reference to this paper should be made as follows: Sharafian, A. and Ghasemi, R. (2017) ‘Stable state dependent Riccati equation neural observer for a class of nonlinear systems’, Int. J. Modelling, Identification and Control, Vol. 28, No. 3, pp.256–270. Biographical notes: Amin Sharafian received his MSc in Control Engineering from the University of Qom in 2016. His research interests include nonlinear control, fuzzy systems, neural networks, fractional calculation, multi-agent systems, etc. Reza Ghasemi received his BSc in Electrical Engineering from the Semnan University in 2000 and MSc and PhD in Control Engineering from the Amirkabir University of Technology, Tehran, Iran in 2004 and 2009, respectively. His research interests include large-scale systems, adaptive control, robust control, nonlinear control, intelligent systems and multi-agent system. He joined the Department of Electrical Engineering, University of Qom, Qom, Iran, where he is currently an Assistant Professor of Electrical Engineering.

1

Introduction

The state estimation of nonlinear systems with uncertainties and unmodelled dynamics has been an active field of research in the past decade (Zhang and Sinha, 2009). This paper investigates a new alternative technique based on combination of state dependent Riccati equation (SDRE) and artificial neural network (ANN) for state estimation of nonlinear systems. SDRE technique is one of the few successful approaches for solving nonlinear optimal controller problem. During the last decade, optimal innovative methodologies burst into the many control fields (Batmani and Khaloozadeh, 2013; Lee and Choi, 2014). such as optimal vehicle control (Alirezaei et al., 2013), satellite attitude control using gas jet and reaction wheels (de Souza and Arena, 2013), optimal control for permanent magnet synchronous motor (Pang and Liu, 2012), optimal

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controller design for flexible robot manipulators (Shawky et al., 2013), a new optimal controller design for on the generalised (n-link) inverted pendulum system (Jadlovska and Sarnovsky, 2013), a new optimal exact input-state feedback linearisation of a nonlinear Maglev system is using state estimator (Benomair and Tokhi, 2016), a swarm intelligent algorithm called gradient-based continuous ant colony optimisation (GCACO) was utilised to design controller in multi-variable nonlinear control systems (Eftekhari and Eftekhari, 2016), a global synchronisation of a stochastic version of coupled map lattices networks through an innovative stochastic adaptive linear quadratic pinning control methodology is derived in Herzallah (2015), an optimal controller design of stochastic dynamical system with delay is proposed in Udom (2015), etc. Meanwhile, hybrid SDRE observers have been implanted in a number of applications (Lee and Choi, 2014) such as exponential

Stable state dependent Riccati equation neural observer for a class of nonlinear systems observer based on SDRE technique which is investigated in Beikzadeh and Taghirad, 2011, 2012). The main advantages of using SDRE technique in observer design are overcoming the singularity problem, accurate flawless estimation and convergence of the error to the neighbourhood of zero in a bounded time (Jin, 2015). SDRE method transforms the nonlinear system into seemingly linear (i.e., pseudo-linear) structure by using state dependent coefficient (SDC) formulation (Khamis and Naidu, 2013; Pakki et al., 2014). Therefore, the nonlinearity of the system can be entirely covered because no approximation is applied. In the process of observer design we need to deal with physical dynamical system problems such as uncertainties and unmodelled dynamic (Misawa et al., 2014; Liu et al., 2013). In order to overcome these problems properly, we need to apply a new hybrid accurate state estimation approach therefore, the combination of SDRE technique and neural network makes a new approach method to cover these uncertainties to have a flawless state estimation of the internal states of a physical system. The ANN is one of the successful artificial intelligence methods that can be used to improve the control process of nonlinear systems with uncertainties (Khoygani and Ghasemi, 2016; Khoygani et al., 2015; Lin, 2015). In the observer design process neural networks are utilised to diagnose the unknown dynamics of the system as the weights of the NN are being adjusted by using adaptation laws in order to obtain the stability of the overall scheme (Wang and Ren, 2016). The NN is trained with the error back-propagation learning algorithm by using a correction term which guarantees the suitable training of the neural network. The combination of SDRE strategy with neural observer properly approximates the measurement noise and allows to adopt a more reliable nominal model. In this paper, the SDRE technique adopted to achieve a reliable observer gain from known nonlinear parts of the system by pseudo-linearising these known part and solving the Riccati equation in every loop. Therefore, the observer gain is a time-varying variable simultaneously; the neural network part of the proposed approach observer estimates the unknown nonlinearities of the system by use of BP algorithm. The combination of these methods makes a new robust and stable observer which is definitely have a better performance compare to SDRE filters or neural observers alone. The important merit which makes this methodology superior in comparison with past literature is the observer gain which adaptively adapted by the SDRE methodology. On the other hand, SDRE observers which mentioned in past studies did not have such proper capability to distinguish the nonlinear uncertainties of the system; therefore, we determined to apply optimal control method in line with neural estimation to decrease the value of the observer gain as much as possible. Moreover, there are adjustable parameters available in order to achieve a proper observer gain based on the complexity and different terms in the actual systems.

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The paper is organised as follows: Section 2 introduces a SDRE-based neural observer on nonlinear system which is trained by back propagation learning algorithm and provided the stability of the observer by Lyapunov theory. Section 3 presents simulation and results and the last section contains a brief conclusions.

2

SDRE-based neural observer

Consider a nonlinear MIMO system as follows: ⎧ x = f ( x) + g ( x, u ) ⎨ ⎩ y = h( x )

(1)

where f(x) is the known smooth function of the system and g(x, u) is unknown smooth nonlinear dynamics of the system. It is assumed that the f(x), h(x) can be transformed into the pseudo-linear structure as follows: ⎧ x = A( x) x + g ( x, u ) ⎨ ⎩ y = C ( x) x

(2)

where A(x) and C(x) are the components of pseudo-linear structure matrices for the observer design. SDC parameterisation is the process of factorising a nonlinear system into a linear-like structure contains SDC matrices. There are two assumptions which determine a nonlinear function always exist in SDC form: 1

f(0) = 0

2

f(.) ∈ C1(ℜn), is a continues nonlinear matrix-valued function.

By invoking a Luenberger observer, the observer model of the system can be defined in this way: ⎪⎧ xˆ = A ( xˆ ) xˆ + gˆ ( xˆ , u ) + L ( xˆ ) ( C ( x) x − C ( xˆ ) xˆ ) ⎨ ⎪⎩ yˆ = C ( xˆ ) xˆ

(3)

where xˆ is the observer state and L( xˆ ) is the time-variant observer gain. The SDRE can be obtained as follows: ˆ ( xˆ ) + P ( xˆ ) AT ( xˆ ) xˆ P ( xˆ ) = A ( xˆ ) xP −1

ˆ ( xˆ ) C ( xˆ ) P ( xˆ ) + Q ( xˆ ) − P ( xˆ ) C ( xˆ ) xR T

(4)

where Q(x) and R(x) are positive definite and positive semi-definite matrices respectively. P( xˆ ) is the solution of Riccati equation and the suboptimal observer gain can be obtained with this formulation: ˆ ( xˆ ) L ( xˆ ) = P ( x)C ( xˆ ) xR T

−1

(5)

where L( xˆ ) would be the optimal observer gain. The process of designing the SDRE observer has been completed. Now we need to combine the NN part of the

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observer to estimate unknown nonlinear part of the physical system. The ANN structure applied in this paper is a single hidden layer neural network which could approximately estimate all of nonlinear functions to any desirable degree of accuracy under certain conditions. The following theorem shows the capability of ANN function approximation. Theorem 1: A feed forward MLP neural network with only one hidden layer with nonlinear activation functions and linear activation functions of the output layer. If the activation functions of the hidden layer satisfy the Riemann integrable condition, then the feed forward neural network could estimate any nonlinear function with desirable accuracy (Pratap and Purwar, 2014). Thus, based on the above theorem, the system could be rewritten as follows: ⎧ x = A( x) x + wT φ ( x, u ) + ε ⎨ ⎩ y = C ( x) x

(6)

the observer for the system mentioned in equation (6) could be rewritten as: ⎧⎪ xˆ = A ( xˆ ) xˆ + wˆ T φ ( xˆ , u ) + L ( xˆ ) ( C ( x) x − C ( xˆ ) xˆ ) ⎨ ⎪⎩ yˆ = C ( xˆ ) xˆ

(7)

In the observer design process, we need to define the error between observer dynamic and the system and try to minimise it. Therefore, the observer error is defined by:

x = x − xˆ

(8)

The observer error is not available in the actual systems because the states of the system are not available but we should coverage it to zero to proof the performance of the proposed observer. The observer error dynamic could be written as equation (9). x = Ac ( x ) x + w T φ ( xˆ , u ) + wT ⎣⎡φ ( x, u ) − φ ( xˆ , u ) ⎦⎤ + ε

(9)

where w = w − wˆ and L( x ) chosen so that Ac ( x ) = A( x ) − L( x )C ( x ) be Hurwitz for all x. To update NN weight matrices, the cost function is considered as follows: J=

1 T y y 2

(10)

where y = y − yˆ. The back propagation algorithm with a modification term is given as: ∂J − ρ y wˆ wˆ = −η ∂wˆ

(11)

where η is the learning rate and ρ is the damping coefficient. Figure 1

Structure of state dependent Riccati equation neural observer (see online version for colours)

Stable state dependent Riccati equation neural observer for a class of nonlinear systems ∂J

According to the chain rule,

∂wˆ

1 2 V ≤ − λmin Q2 x + ⎡⎣ x p ( w φ ( xˆ, u ) + ω ) 2 2 +φM w x l1 + ⎡⎣ wM w − w ⎤⎦ ρ C ( xˆ ) x

can be expressed as

follow: ∂gˆ ( xˆ, u ) ∂J ∂J ∂y ∂x . . . = = ∂wˆ ∂y ∂x ∂gˆ ( xˆ , u ) ∂wˆ y C ( xˆ ) T

∂x φ ( xˆ, u ) gˆ ( xˆ, u )

(12)

(

∂x ∂x −1 × −I → ≈ Ac ( xˆ ) ∂gˆ ( xˆ , u ) ∂gˆ ( xˆ , u )

(32)

Thus, the update rule for the modular NN weight matrices can be rewritten as mentioned in equation (14).

)

−1 wˆ = −ηφ ( xˆ , u ) y T C ( xˆ ) Ac ( xˆ ) − ρ y wˆ

+ w

(16)

where Q2 is a positive definite function too. By differentiating the Lyapunov function, we have: 1 1 V = xT px + xT px + tr ( w T w ) 2 2

(17)

After some mathematical manipulation based on the equations (9) and (14), the above equation is rewritten as: 1 V = − xT Q2 x + xT p ⎡⎣ w T φ ( xˆ, u ) + ω(t ) ⎤⎦ 2 +tr ( w T ηφ ( xˆ , u ) ( y T CAc ( xˆ ) −1 ) ) + w T ρ y wˆ

(18)

where ω(t ) = wT (φ ( x, u ) − φ ( xˆ , u ) ) + ε ( x). Then by applying the following inequalities: tr ( w T ( w − w ) ≤ wM ) w − w

2

tr ( w T φ ( xˆ, u ) xT l1 ) ≤ φM w x l1

|| l1 || || P || φM || l1 || + wM ρ || C ( xˆ ) || and N 2 = , 2 2( ρ || C ( xˆ ) || − N12 )

)

− ( ρ C ( xˆ ) − N12 )( N 2 − w

(19)

Using the equation (19), we can rewrite the equation mentioned in (18) as the follows:

(22)

2 ) − ( N1 || w ||)2 ⎤⎦

x

The V is negative as long as the following compact set: ⎧ x | x > ⎫ ⎪ ⎪ 2 ⎡ Ωc = ⎪⎪ || p || ω + ρ C ( xˆ ) − N12 N 22 ⎤ ;⎪⎪ ⎣ ⎦ λ Q ⎨ min ⎬ ⎪ ⎪ 2 ⎪∀ρ ≥ N1 ⎪ ⎪⎩ C ( xˆ ) ⎭⎪

(

where p is a positive definite matrix satisfying the following formulation: Ac ( xˆ ) p + pAc ( xˆ ) = −Q2

)

2 p φM + φM l1 + wM ρ C ( xˆ ) − N12 w ⎤ x ⎦

(

(15)

T

(21)

1 2 V ≤ − λmin Q2 x 2 + x ⎡|| p || ω + ρ C ( xˆ ) − N12 N 22 ⎣

Consider the following Lyapunov candidate function: 1 T 1 x px + tr ( w T w ) 2 2

)

we have:

Proof:

V=

(

where N1 =

(14)

Theorem 2: Consider the system that mentioned in equation (6), the stability of the closed-loop system and convergence of the states of the observer mention in equation (7) to zero with update laws in equation (14) to neighbourhood of zero is guaranteed. Furthermore, the boundedness of the all signals involved in closed-loop system is guaranteed.

(20)

where l1 = ηC T CAc−1 and the above equation can be expressed as: 1 2 V ≤ − λmin Q2 x 2 + ⎡ p w − w ρ C ( xˆ ) − N12 ⎣

∂x = ( A ( xˆ ) − L ( xˆ ) C ( xˆ ) ) ∂gˆ ( xˆ, u )

(

259

)

(23)

This shows the uniform ultimately boundedness of observer error and the signals involved in closed-loop system. The proof is complete.

3

Simulation and results

In this section, we investigate the performance of our approach methodology on an example system and on two chaotic systems to show the excellent ability of state estimation of the methodology. The results are as follows.

3.1 Case 1 Consider the following nonlinear system. ⎧ x1 = x2 − x1 ⎪ 3 ⎨ x2 = − x1 − x2 sin ( x2 ) + g ( x, t ) + u ⎪y = x 1 ⎩

(24)

where g ( x, t ) = 3sin( x23 ) + 12 cos(t ) + x12 is the unknown part of the nonlinear system and u = sin(t). The certain nonlinear part equations of the system can be transformed into the pseudo-linear structure as follows:

260


⎛ −1 A( x) = ⎜ 2 ⎝ − x1

1 ⎞ ⎟ − sin ( x2 ) ⎠

(25)

and C ( x) = (1 0)

(26)

The parameters Q and R can be used as design parameters to penalise the state variables and the observer gain signals. By choosing some values for R and Q, the state estimation results would be investigated as follows. Consider that 0 ⎞ ⎛0 R = ( 5 × 103 ) , Q = ⎜ ⎟ 0 1 × 105 ⎠ ⎝

Figure 2 illustrates the state x1 which is demonstrated in the output of the system, and we can see by choosing a large value of R the estimation error converges to zero after a limited time. Figure 3 shows the second state of the system which is not demonstrated in the output of the system. As we can see, due to large value of R, bounding variance of L is decreased and consequently the state estimation accuracy is decreased. Therefore, because of the bounded value of L the estimation error cannot converge to zero and the estimation is weak. By decreasing the value of R the estimation accuracy increases as shown in Figures 5 to 7.

Figure 2

State x1 and SDRE estimation (R = 5,000) (see online version for colours)

Figure 3


Stable state dependent Riccati equation neural observer for a class of nonlinear systems Figure 4

State x2 estimation error (R = 5,000) (see online version for colours)

Figure 5


Figure 6


261

262


Figure 7

State x2 estimation error (R = 1,000) (see online version for colours)

Figure 8

State x1 and SDRE estimation (R = 1) (see online version for colours)

Figure 9

State x2 and SDRE estimation (R = 1) (see online version for colours)


State x2 estimation error (R = 1) (see online version for colours)

Figure 11

Duffing system state x1 and SDRE estimation (R = 8,000) (see online version for colours)

Figure 12

Duffing system state x2 and SDRE estimation (R = 8,000) (see online version for colours)

263

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Figure 13

Duffing system state x2 estimation error (R = 8,000) (see online version for colours)

Figure 14

Duffing system state x1 and SDRE estimation (R = 5) (see online version for colours)

Figure 15

Duffing system state x2 and SDRE estimation (R = 5) (see online version for colours)


265

Duffing system state x2 estimation error (R = 5) (see online version for colours)

In Figures 5, 6 and 7, decreasing the value of R makes the estimation error converge to zero faster; eventually the small value of R leads to larger value for L therefore after convergence we have a more bounded error. By comparing signal of the estimation error which is illustrated in Figures 4 and 7, we expect that more decreasing the value of R leads to more accurate and reliable estimation and in the following we can see by choosing a very small value of R the estimations are still improved. The results are illustrated in Figures 8, 9 and 10. In Figure 8, 9 and 10, it obvious that more decreasing the value of the R would deliver the most accurate estimation and with the minimum error which is converging to zero very fast.

3.2 Case 2 In this case, we applied our new methodology to a chaotic system to investigate the performance of the observer against chaotic behaviour of such systems. Consider Duffing chaotic nonlinear system with uncertainties as shown in Figures 11 and 12. ⎧ x1 = x2 ⎪ 3 ⎨ x2 = −0.1x2 + x1 + g (t ) + u ⎪y = x 1 ⎩

(27)

where g(t) = 12cos(t) is the unknown part of the system and the pseudo-linear structure is as follows: ⎛0 A( x) = ⎜ 2 ⎝ x1

1 ⎞ ⎟ −0.1⎠

(28)

By choosing a large value for R the state estimation results as shown in Figures 11 to 13. Eventually by choosing a very small value of R the estimations are improved. The simulation results show that proposed observer could estimate states of a mentioned system with a

satisfactory accuracy. By choosing a proper value for R and Q, the appropriate state estimation with limit boundedness is derived.

3.3 Case 3 In this case, we investigated another more complicated MIMO chaotic system to fully examine the performance of the proposed observer. Here, we decided to choose a constant value for R and change the value of Q to find an optimum observer gain which could track the actual system signal rapidly, therefore we should choose a small value of Q and increase it in every step to find the optimum gain which make all states tracked exactly by the proposed observer. Chen system is described by the following dynamic: ⎧ x1 = 40 ( x2 − x1 ) ⎪ ⎪ x2 = −12 x1 + g1 (t ) ⎨ ⎪ x3 = x1 x3 + g 2 (t ) + u ⎪⎩ y = Cx

(29)

⎡1 0 0 ⎤ C=⎢ ⎥ ⎣0 1 0 ⎦ and g1(t) = –x1x3 + 28x2, g2(t) = where –3x3, u = 10sin(t) which are assumed to be uncertain in the observer design. Therefore, the SDC parametrisation become as follows: ⎛ −40 40 0 ⎞ ⎜ ⎟ A( x ) = ⎜ −12 0 0 ⎟ ⎜ x 0 0 ⎟⎠ ⎝ 3

(30)

Or ⎛ −40 40 0 ⎞ ⎜ ⎟ A( x ) = ⎜ −12 0 0 ⎟ ⎜ 0 0 x1 ⎟⎠ ⎝

(31)

266


which make no difference in the final results but we should adopt second one to satisfy the observability rules. By choosing the following value for Q and R: 0 ⎞ ⎛ 400 0 ⎛ 600 0 ⎞ ⎜ ⎟ R=⎜ ⎟ , Q = ⎜ 0 400 0 ⎟ 0 600 ⎝ ⎠ ⎜ 0 0 400 ⎟⎠ ⎝

The results become as shown in Figures 17 to 19. As we could see in Figures 17, 18 and 19 by choosing a small value of Q there is a transient phase until the observer tracks the output signal so we should increase the value of Q to achieve a better estimation. Figure 20 illustrates the estimation error which converges to the neighbourhood of zero after 2 seconds. Now we should increase the value of Q to omit the transient phase:

⎛ 3 × 104 ⎛ 600 0 ⎞ ⎜ R=⎜ ⎟, Q = ⎜ 0 0 600 ⎝ ⎠ ⎜ 0 ⎝

0 3 × 10 0

⎞ ⎟ ⎟ 3 × 104 ⎟⎠ 0

4

0

In Figures 21 to 23, we can see the result of increment. Figures 21, 22 and 23 show that the observer could estimate the states of the system faster and with minimum estimation error. As it is obvious in Figure 24 by increasing the value of Q the state estimation error was reduced in the first seconds of estimation and also the steady state error decreased. Therefore, we could find a proper observer gain for each system by taking advantage of the adjustable parameters in the SDRE estimation.

Figure 17

Chen system state x1 and SDRE estimation (Q = 400) (see online version for colours)

Figure 18




Figure 20

Chen system state x3 estimation error (Q = 400) (see online version for colours)

Figure 21

Chen system state x1 and SDRE estimation (Q = 30,000) (see online version for colours)

267

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Figure 22


Figure 23


Figure 24

Chen system state x3 estimation error (Q = 30,000) (see online version for colours)

Stable state dependent Riccati equation neural observer for a class of nonlinear systems

4

Conclusions

In this paper, a new approach observer is applied on nonlinear systems with unknown nonlinear parts. The lack of hybrid observers based on state dependent Riccati equation in the past literature is ignitable, therefore the combination of the SDRE technique and the neural network has been applied in this paper to develop a new approach for designing optimal observers that has a better performance than the typical NN observer. The uniform ultimate boundedness of the observer error is guaranteed by Lyapunov theory. In the simulation results, three systems are adopted to investigate the performance of the observer and, as shown in the figures, we can control the gain of the observer by two adjustable terms in order to have a better estimation in different situations and different dynamical models which makes this method work properly for wider range of dynamical models. The stability of the closed-loop system, the robustness of the proposed observer against uncertainties and overcoming of the singularity problem are the main merits of the proposed observer design procedure. Moreover, we could apply the proposed observer to estimate the infected cells in chemotherapy.

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