Robust Adaptive Control Applied to Chua's Circuit

Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011

Robust Adaptive Control Applied to Chua’s Circuit Samaherni M. Dias ∗ Allan de M. Martins ∗ Aldayr D. de Araujo ∗ Kurios Queiroz ∗ ∗

Federal University of Rio Grande do Norte, Department of Electrical Engineering, Laboratory of Automation, Control and Instrumentation (LACI), Natal, RN, Brazil (e-mail:[email protected])

Abstract: Many practical systems (for example: robotic systems, power system and electronic circuits) are multiple-input multiple-output nonlinear systems and some of them have coupled relations between inputs and outputs. Besides all that, these systems can suffer from plant uncertainties and external disturbances. Any control techniques to be applied to these systems are complex. This work proposes a new control structure, based on the union between the variable structure model reference adaptive control and a decouple left-inverse technique, to transform the nonlinear multiple-inputs multiple-outputs system into a number of single-input single-output linear systems. In that case each input affects only one output and with a desired closed-loop performance. The proposed structure uses only input/output measurements, improves the transient performance (reducing the output “chattering”) and it is robust to parametric uncertainties and disturbances. All these features are demonstrated by simulation results of a simple electronic circuit that exhibits chaotic behavior (Chua’s circuit). The proposed structure may be used to smooth the control signal, thus reducing the chattering in the output signal. Keywords: Robust control; Model reference adaptive control; Variable structure control; Decoupled subsystems; Chattering. 1. INTRODUCTION Nowadays, there has been an increasing interest in applying control techniques for industrial processes. However, many of these processes are nonlinear MIMO (MultipleInput Multiple-Output) systems and some of them have coupled relations between inputs and outputs. Besides all that, these systems can suffer from plant uncertainties and external disturbances. Any control techniques to be applied to these systems are complex. This work proposes to decouple the nonlinear MIMO system to get a number of SISO (Single-Input Single-Output) linear systems, in which each input affects only one output and with a desired closed-loop performance. Some works in this area can be highlighted for their contributions, such as [6], in which Hirschorn proved the sufficient condition of the left-inverse existence for a class of nonlinear systems (minimum phase system), in Singh [13] the algorithm of constructing inverse system proposed by Hirschorn was modified and gave a new invertibility condition for a class of systems which do not satisfy Hirschorn’s invertibility condition, and Li et al [10] generalised the inverse system method to the more generic nonlinear system and gave the sufficient and necessary invertibility condition. Nowadays, some works are using artificial neural networks to approximate a proper inverse system [3, 14]. Recently, the application of nonlinear decoupling control methods has been proposed [5, 1, 11, 12, 14, 4]. Many of 978-3-902661-93-7/11/$20.00 © 2011 IFAC

these applications are directed to induction motors and robotic systems. The application presented here is related to electronic circuits. We are using a modified algorithm of constructing inverse system, proposed by Hirschorn, associated with a sliding mode control technique to decouple a modified Chua’s circuit, which is a simple electronic circuit that exhibits chaotic behavior. The modified Chua’s circuit is very sensitive to variations in its components and has a strong coupling between inputs and outputs. These are the reasons to choose the Chua’s circuit to test the proposed technique. The sliding mode control technique used in this work is the Variable Structure Model Reference Adaptive Control (VS-MRAC). This strategy offered remarkable stability and performance robustness properties with respect to parametric uncertainties, unmodelled dynamics and external disturbances [2], as well as, fast transient performance. The VS-MRAC for linear plants with relative degree one was proposed in [9], and then extended in [7], for the general case. The application of switching techniques in electronic circuits is not new, the most successful application is the switched-mode power supply (SMPS). Initially, we will decouple, using a proposed modified algorithm, a nonlinear MIMO system (Chua’s circuit) into two linear SISO systems, and then for each decoupled SISO system we will apply a VS-MRAC controller.

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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

 m X   x˙ = A(x) + ui bi (x);

2. NONLINEAR INVERTIBILITY Based on algorithm of constructing inverse system proposed by Hirschorn [6], which constructs a sequence of systems by changing the output maps until one can solve for u in terms of y, its derivatives, and x. It is then possible to write down a second nonlinear system which acts as a left-inverse for the original system. We will assume that the reader is familiar with the notation and results on nonlinear invertibility in [6]. Consider the system  m X   x˙ = A(x) + ui bi (x); i=1   y = C(x)

x ∈ M,

(1)

where A, b1 , . . . , bm ∈ V (M :→ M ) and C : M → Rl is a real analytic mapping. Then dy = y (1) = Ac(x) + D(x)u (2) dt m where u denote the vector in < whose components are u1 , . . . , um and D(x) = [b1 c(x) b2 c(x) · · · bm c(x)] an l ×m matrix for each x ∈ M . Now, consider the system 1  m X   x˙ = A(x) + ui bi (x);

x ∈ M1 ,

i=1

  where

(3)

z1 = C1 (x) + D1 (x)u dy dt C1 = R0 (x)Ac(x) z1 = R0 (x)

System J  m X   x˙ = A(x) + ui bi (x); i=1

zα = Cα (x) + Dα (x)u and by construction   Dα1 (x) Dα (x) = 0

x ∈ MJ ,

(6)

 

zJ = CJ (x) + DJ (x)u where MJ is an open dense submanifold of M , CJ (x) and DJ (x) are l × l and l × m matrices, respectively, whose entries are real analytic functions on MJ , and   DJ1 (x) DJ (x) = (7) 0 with DJ1 (x) a rJ × m matrix of rank rJ for all x ∈ MJ . By construction 0 6 r1 6 r2 6 r3 6 . . . 6 m (8) where m is the number of inputs. Thus, there exists a least positive integer J such that rJ is maximal.

(9)

(10)

where for all x ∈ Mα , Dα1 (x) is an rα × m matrix of rank rα , since α < ∞, rα = m, and Dα1 (x) is an invertible m×m matrix. Let z α and cα denote the first m components of zα and cα , respectively. Then z α = cα (x) + Dα1 (x)u. (11) If x0 ∈ Mα then exist an m × αl matrix Hα (x) whose entries are real analytic functions on Mα such that  (1)  y (t)   z α (t) = Hα (x(t))  ...  (12) y (α) (t) and the system  b x) + B(b b x)b x b˙ = A(b u; x b0 = x0 ∈ Mα , b b x)b yb = C(b x) + D(b u with state manifold Mα −1 b x) = A(b x) . . . bm (b x)] Dα1 A(b x) − [b1 (b (b x)cα (b x) −1 b x) = [b1 (b x) . . . bm (b x)] Dα1 (b B(b x)Hα (b x) b x) = −D−1 (b C(b x) α1 x)cα (b −1 b D(b x) = Dα1 (b x)Hα (b x) acts as a left-inverse for the system (1).

(4)

D1 = R0 (x)D(x) R0 (x) is a matrix with the property that reorders the rows of D(x) and   D11 (x) R0 (x)D(x) = (5) 0 where D11 (x) is a r1 × m matrix of rank r1 for all x ∈ M1 and r1 =maxx∈M {rank D(x)} is called the invertibility index of system (1).

x ∈ Mα ,

i=1

 

(13)

(14)

3. VS-MRAC CONTROLLER The VS-MRAC (Figure 1) was proposed in [9, 7, 8]. The objective of VS-MRAC is to find the feedback control law that changes the structure and dynamics of the plant so that its inputs/outputs properties are exactly the same as those of the reference model. Consider a SISO linear time-invariant plant with strictly proper transfer function kp np (s) = W (s) = kp , dp (s) s + ap input u and output y. The model reference is characterized by the strictly proper transfer function nm (s) km M (s) = km = , dm (s) s + am input yr and output ym. The purpose is to find a control law u(t), using only the plant input and output measurements, such that the output error e0 = y − ym (15) tends to zero asymptotically for arbitrary initial conditions and arbitrary piece-wise continuous uniformly bounded reference signals yr (t). Let us consider where

Based on [6], suppose that a system of the form (1) has relative order α < ∞. Then the αth system will be 5844

θ1∗ =

u = θ∗T ω ap − am kp

, θ2∗ =

(16) km kp

(17)

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

Model M (s)

yr

ur1

ym

θ2 = −θ 2 sgn(e0 · yr ) urξ -

θ2

e0

θ2

+

u

Plant W (s)

u1 Left . . Inverse . u System m

y1 . . .

MIMO System

yl

ur1

⇒

urξ

s−α . . . s−α

y1

yξ

Relay

+ +

. . .

Figure 2. Block diagram of Hirschorn’s inverse system method

y θ1 Relay

with relative degree α ur1

θ1 θ1 = −θ 1 sgn(e0 · y)

urξ

Figure 1. Block diagram of variable structure model reference adaptive controller (VS-MRAC) as the control signal to the plant (W (s)) matches the model reference (M (s)) exactly, i.e., the transfer function of the closed-loop plant, from yr to y is M(s). Of course, θ∗T can only be known if W (s) is known. When this is not the case the control signal is u = θT ω, (18) where θT = [θ1 θ2 ] is the vector of adaptive parameters (under some signal richness condition θ → θ∗ ) and T

ω = [y yr ] , is defined as “regressor” vector.

(19)

The following usual assumptions are made: 1. the relative degree n∗ of the plant W (s) is known and the model reference M (s) has the same relative degree; 2. only the plant input and output are used to generate the control signal u; 3. the order of the plant is known, say n, i.e., dp (s) is monic of order n; 4. the plant and the model are supposed to be completely observable and controllable (the pairs of monic polynomials (np , dp ) and (nm , dm ) are coprime); 5. the signs of kp and km the “high frequency gains”, are assumed to be the same; 6. W (s) is minimum phase. Thus, the parameter update law is θi = −θi sgn(e0 ωi ) where θi > |θi∗ |, i = 1, 2

. . .

u1 Left . . Inverse . System um

y1 . . .

MIMO System

yl

ur1

⇒

urξ

W1 (s) . . . Wm (s)

y1

yξ

Figure 3. Block diagram of the modified Hirschorn’s inverse system method The modified Hirschorn’s inverse system method will decouple the nonlinear MIMO system to get a number of SISO linear systems with Wi (s) as linear decoupled transfer functions. But, when there is parameter uncertainty in nonlinear MIMO system, the linear decoupled functions can be interpreted as (see Figure 4) yi = W xi (s)(uri + di ) (21) where di = f (u1 , · · · , um ) (22) is an input disturbance, W xi (s) = Wi (s) + 4κ and 4κ is an unmodeled dynamic.

(23)

di +

uri

W xi (s)

yi

+

Figure 4. Linear decoupled functions when there is parameter uncertainty in nonlinear MIMO system To ensure that modified Hirschorn’s inverse system method will decouple the nonlinear MIMO system, we are using a VS-MRAC controller (Vi ), to each decoupled linear system (see Figure 5). The VS-MRAC controller offers remarkable stability and performance robustness properties with respect to parametric uncertainties, unmodelled dynamics and external disturbances. di

4. CONTROLLER STRUCTURE

yri

Vi

+

uri

W xi (s)

yi

+

This work proposes, using a modified Hirschorn’s inverse system method, to decouple the nonlinear MIMO system to get a number of SISO linear systems (see Figures 2 and 3), in which each input affects only one output. The Figure 2 presents a block diagram of Hirschorn’s inverse system method. It is important to observe that a number of SISO linear systems is ξ, where ξ ≤ m. The modified Hirschorn’s inverse system method proposed b x) (system (see Figure 3) here is obtained by change A(b 13). The main idea behind this modification is change the linear decoupled transfer functions of s−α to bwn−α sn−α + · · · + bw1 s + bw0 (20) W (s) = n s + awn−1 sn−1 + · · · + aw1 s + aw0

Figure 5. VS-MRAC controller applied to the system obtained using the modified Hirschorn’s inverse system method The proposed controller decouples the MIMO linear system into that of a number of SISO linear systems with transfer function given by the model reference M (s) of VS-MRAC controller (see Figure 6). 5. SIMULATION RESULTS This section presents an example which highlights the performance of the proposed controller. Therefore, the

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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

yri

Mi (s)

where

yi

Figure 6. Set of decoupled linear transfer functions obtained using proposed controller proposed controller is applied to a modified Chua’s circuit, which is very sensitive to variations in its components and has a strong coupling between inputs and outputs. The modified Chua’s circuit in state-space is given by ( y˙ 1 = k1 (y2 − y1 ) − k2 g(y1 ) + u1 y˙ 2 = k3 (y1 − y2 ) + k4 y3 + u2 (24) y˙ 3 = −k5 y2 + u2 where g(x) = Gb x + 0.5(Ga − Gb )[|x + Bp | − |x − Bp |] (25) is a nonlinear function and Ga , Gb , Bp , ki (i = 1, . . . , 5) are auxiliary constants depending on physical components of the circuit. 5.1 Decoupling the modified Chua’s circuit The aim is to decouple the nonlinear MIMO system (24) into two linear SISO systems (see Figure 7), one linear system to y1 and one to y2 , where y1 and y2 represent the voltage in two different capacitors of the circuit. ur1

ur2

u1 Left Modified Inverse Chua’s System u2 Circuit

y1

y2

ur1

⇒

ur2

W1 (s)

W2 (s)

y1

y2

Figure 7. Block diagram of the modified Hirschorn’s inverse system method The left-inverse system for (24), using the method presented in section 2, is given by yb˙ 1 = yb1 + ur1 (26) yb˙ 2 = yb2 + ur2 yb˙ = −k5 yb2 3

with

u1 = −k1 (b y2 − yb1 ) + k2 g(b y1 ) + ur1 u2 = −k3 (b y1 − yb2 ) − k4 yb3 + ur2

(27)

If we change (26) to yb˙ 1 = −km yb1 + km ur1 (28) yb˙ 2 = −km yb2 + km ur2 ˙yb = −k5 yb2 3 where km is defined by the model reference of VS-MRAC controller, the modified Hirschorn’s inverse system is obtained. 5.2 Design of VS-MRAC controllers First of all, it is necessary to define the reference models (M1 (s), M2 (s)) ymi km Mi (s) = = yri s + km The second step is to define the control law to each decoupled linear system T ur1 = θv1 ω1 (29) T ur2 = θv2 ω2

ωiT = [yi yri ]

T θvi = [θi 1 θi 2 ] with i = 1, 2. Thus, the parameter update law is

θ1 1 θ1 2 θ2 1 θ2 2

= = = =

−θ1 1 sgn(e0 1 y1 ) −θ1 2 sgn(e0 1 yr1 ) −θ2 1 sgn(e0 2 y2 ) −θ2 2 sgn(e0 2 yr2 )

(30)

(31)

where e0 i = yi − ymi 5.3 Simulations Consider the modified Chua’s circuit (system 24) with initial conditions # # " " 0.15264 x(0) y(0) = −0.02281 , (32) 0.38127 z(0) and  k1 = 7   (   k2 = 10 Bp = 1 k3 = 0.35 Ga = 4 (33)   k = 0.5 G = 0.1 4 b   k5 = 7 In the design of VS-MRAC controller, the model reference is chosen km = 20 and θ1 1 = 0.3375 θ1 2 = 1.4250 (34) θ2 1 = 0.2756 θ2 2 = 0.3937 Another important consideration is that all simulations has 400s and at t > 250s the parameters values of the system will change to  k1 = 10     k2 = 12.5 k3 = 0.363 (35)   k   4 = 0.454 k5 = 8.139 in the simulations of Figures (9-11). The chaotic behavior of modified Chua’s circuit is shown in Figure 8. The Figures (e-f) presents the behavior of system (24) when the inputs u1 and u2 has form given by the Figures 8(c-d), respectively. The simulation of the Figure 8 (a-b) shows that the behavior of the modified Chua’s circuit remains chaotic despite input signals different from zero. Next simulation (Figure 9) presents the behavior of system (24) using (28-27) as left-inverse system. In this simulation it is important to note that any constant value to ur2 leads the system to instability 1 . The behavior of system using the proposed left-inverse system (Figure 9) is oscillatory and presents output error. When the system parameters change (t > 250s) the output 1

It is important to point out that the left-inverse should have decoupled the system perfectly. That was not the case due to rounding in the parameters and the sensitivity of the chaotic system.

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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

The Figure 10 shows the behavior of system (24) using (28-27) as left-inverse system and designed VS-MRAC controllers (section 5.2).

(a) y3 × y1

(c) t × u1

(e) t × y1

(b) y1 × y2

(a) y3 × y1 and yr3 × yr1

(b) y1 × y2 and yr1 × yr2

(c) t × (y1 and yr1 )

(d) t × (y2 and yr2 )

(e) t × u1

(f) t × u2

(d) t × u2

(f) t × y2

Figure 8. Simulation of modified Chua’s circuit, with different values to the inputs u1 and u2

Figure 10. Simulation of modified Chua’s circuit with leftinverse system and VS-MRAC controllers In this simulation (Figure 10) the focus is the behavior of the proposed controller, which has a fast transient and a small “chattering” 2 on the output signal. Another aspect is that proposed controller is robust to parametric uncertainties and input disturbances. (a) t × (y1 and ur1 )

(b) t × (y2 and ur2 )

Finally, the Figure 11 shows the behavior of system (24) without left-inverse system and using the VS-MRAC controllers (section 5.2) with θ1 1 = 12.83 θ1 2 = 54.15 (36) θ2 1 = 6.89 θ2 2 = 9.84 which are the smaller values to VS-MRAC controller ensure that the output error e0 tends to zero asymptotically.

(c) t × u1

(d) t × u2

Figure 9. Simulation of modified Chua’s circuit with leftinverse system error increases. Despite inverse system, there is a coupling between the inputs and outputs of the system. A strong coupling between ur2 and y1 , which makes the system unstable, and a weak coupling between ur1 and y2 .

According to (21), any adaptive controller with a fast transient can be used to decouple a nonlinear MIMO system into that of a number of SISO linear systems. The simulation of Figure 11 was presented to show that 2

“Chattering” phenomenon are high frequency oscillations about the switching line in sliding mode, which occurs due to the existence of non-ideal relays, having hysteresis effects and finite time delays in switching of the output.

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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

In future works, the stability analysis, applications on industrial environments will be discussed. REFERENCES [1]

(a) y3 × y1 and yr3 × yr1

[2]

(b) y1 × y2 and yr1 × yr2

[3]

[4] (c) t × (y1 and yr1 )

(d) t × (y2 and yr2 )

[5]

[6] (e) t × u1

(f) t × u2

[7] Figure 11. Simulation of modified Chua’s circuit without left-inverse system and using designed VS-MRAC controllers the proposed controller will improve system performance by reducing the output “chattering”. The VS-MRAC controller used was designed to have the smallest possible output chattering. Although the VS-MRAC controller used was designed to have the smallest possible output “chattering”, the proposed controller has a lower output “chattering”. This simulation shows that the application of the inverse system smoothes the control signal and thereby reduces the effect of “chattering” in the output.

[8]

[9]

[10]

[11]

6. CONCLUSION In this work, a new control structure, based on the union between VS-MRAC controllers and the technique of leftinverse system, was proposed. This structure uses only input/output measurements, improves the transient performance (reducing the output “chattering”) and it is robust to parametric uncertainties and disturbances. All these features are demonstrated by simulation results of a simple electronic circuit that exhibits chaotic behavior (Chua’s circuit). This structure may be used for SISO systems, linear or nonlinear, to smooth the VS-MRAC control signal, thus reducing the chattering in the output signal. Finally, this paper presented an application in electronic circuits, which can be expanded to other electronic circuits as switched-mode power supply, modulators, digital-toanalog converter, etc.

[12]

[13] [14]

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