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Robust Adaptive Control Applied to

Chuas Circuit

Samaherni M. Dias Allan de M. Martins

Aldayr D. de Araujo Kurios Queiroz

Federal University of Rio Grande do Norte, Department of ElectricalEngineering, 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 electroniccircuits) are multiple-input multiple-output nonlinear systems and some of them have coupledrelations between inputs and outputs. Besides all that, these systems can suffer from plantuncertainties and external disturbances. Any control techniques to be applied to these systemsare 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, totransform the nonlinear multiple-inputs multiple-outputs system into a number of single-inputsingle-output linear systems. In that case each input affects only one output and with a desiredclosed-loop performance.The proposed structure uses only input/output measurements, improves the transient perfor-mance (reducing the output chattering) and it is robust to parametric uncertainties anddisturbances. All these features are demonstrated by simulation results of a simple electroniccircuit that exhibits chaotic behavior (Chuas circuit). The proposed structure may be used tosmooth 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 apply-ing control techniques for industrial processes. However,many of these processes are nonlinear MIMO (Multiple-Input Multiple-Output) systems and some of them havecoupled relations between inputs and outputs. Besides allthat, these systems can suffer from plant uncertaintiesand external disturbances. Any control techniques to beapplied to these systems are complex. This work proposesto decouple the nonlinear MIMO system to get a numberof SISO (Single-Input Single-Output) linear systems, inwhich each input affects only one output and with a desiredclosed-loop performance.

Some works in this area can be highlighted for theircontributions, such as [6], in which Hirschorn provedthe sufficient condition of the left-inverse existence for aclass of nonlinear systems (minimum phase system), inSingh [13] the algorithm of constructing inverse systemproposed by Hirschorn was modified and gave a newinvertibility condition for a class of systems which do notsatisfy Hirschorns invertibility condition, and Li et al [10]generalised the inverse system method to the more genericnonlinear system and gave the sufficient and necessaryinvertibility condition. Nowadays, some works are usingartificial neural networks to approximate a proper inversesystem [3, 14].

Recently, the application of nonlinear decoupling controlmethods has been proposed [5, 1, 11, 12, 14, 4]. Many of

these applications are directed to induction motors androbotic systems. The application presented here is relatedto electronic circuits.

We are using a modified algorithm of constructing in-verse system, proposed by Hirschorn, associated with asliding mode control technique to decouple a modifiedChuas circuit, which is a simple electronic circuit thatexhibits chaotic behavior. The modified Chuas circuit isvery sensitive to variations in its components and has astrong coupling between inputs and outputs. These are thereasons to choose the Chuas circuit to test the proposedtechnique.

The sliding mode control technique used in this work is

the Variable Structure Model Reference Adaptive Control(VS-MRAC). This strategy offered remarkable stabilityand performance robustness properties with respect toparametric uncertainties, unmodelled dynamics and exter-nal disturbances [2], as well as, fast transient performance.The VS-MRAC for linear plants with relative degree onewas proposed in [9], and then extended in [7], for thegeneral case. The application of switching techniques inelectronic circuits is not new, the most successful applica-tion is the switched-mode power supply (SMPS).

Initially, we will decouple, using a proposed modifiedalgorithm, a nonlinear MIMO system (Chuas circuit) intotwo linear SISO systems, and then for each decoupled SISO

system we will apply a VS-MRAC controller.

Preprints of the 18th IFAC World Congress

Milano (Italy) August 28 - September 2, 2011

Copyright by the

International Federation of Automatic Control (IFAC)

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2. NONLINEAR INVERTIBILITY

Based on algorithm of constructing inverse system pro-posed by Hirschorn [6], which constructs a sequence ofsystems by changing the output maps until one can solvefor u in terms ofy, its derivatives, and x. It is then possibleto write down a second nonlinear system which acts as aleft-inverse for the original system. We will assume thatthe reader is familiar with the notation and results onnonlinear invertibility in [6].

Consider the system x = A(x) +mi=1

uibi(x); x M,

y = C(x)

(1)

where A, b1, . . . , bm V(M : M) and C : M Rl is a

real analytic mapping. Then

dy

dt= y(1) = Ac(x) + D(x)u (2)

where u denote the vector in m whose components areu1, . . . , um and D(x) = [b1c(x) b2c(x) bmc(x)] an lmmatrix for each x M.

Now, consider the system 1 x = A(x) +mi=1

uibi(x); x M1,

z1 = C1(x) + D1(x)u

(3)

where

z1 = R0(x)dy

dt

C1 = R0(x)Ac(x)

D1 = R0(x)D(x)

(4)

R0(x) is a matrix with the property that reorders the rowsof D(x) and

R0(x)D(x) =

D11(x)

0

(5)

where D11(x) is a r1 m matrix of rank r1 for all x M1and r1 =maxxM{rank D(x)} is called the invertibilityindex of system (1).

System J

x = A(x) +

m

i=1uibi(x); x MJ,

zJ = CJ(x) + DJ(x)u

(6)

where MJ is an open dense submanifold of M, CJ(x) andDJ(x) are l l and l m matrices, respectively, whoseentries are real analytic functions on MJ, and

DJ(x) =

DJ1(x)

0

(7)

with DJ1(x) a rJm matrix of rank rJ for all x MJ.

By construction

0 r1 r2 r3 . . . m (8)

where m is the number of inputs. Thus, there exists a leastpositive integer J such that rJ is maximal.

Based on [6], suppose that a system of the form (1) hasrelative order < . Then the th system will be

x = A(x) +mi=1

uibi(x); x M,

z = C(x) + D(x)u

(9)

and by construction

D(x) = D1(x)0 (10)where for all x M, D1(x) is an r m matrix of rankr, since < , r = m, and D1(x) is an invertiblemm matrix. Let z and c denote the first m componentsof z and c, respectively. Then

z = c(x) + D1(x)u. (11)

If x0 M then exist an m l matrix H(x) whoseentries are real analytic functions on M such that

z(t) = H(x(t))

y(1)(t)

...

y()(t)

(12)

and the systemx = A(x) + B(x)u; x0 = x0 M,y = C(x) + D(x)u (13)

with state manifold MA(x) = A(x) [b1(x) . . . bm(x)] D11 (x)c(x)B(x) = [b1(x) . . . bm(x)] D11 (x)H(x)C(x) = D11 (x)c(x)D(x) = D11 (x)H(x)(14)

acts as a left-inverse for the system (1).

3. VS-MRAC CONTROLLER

The VS-MRAC (Figure 1) was proposed in [9, 7, 8]. Theobjective of VS-MRAC is to find the feedback control lawthat changes the structure and dynamics of the plant sothat its inputs/outputs properties are exactly the same asthose of the reference model.

Consider a SISO linear time-invariant plant with strictlyproper transfer function

W(s) = kpnp(s)

dp(s)=

kp

s + ap,

input u and output y. The model reference is characterizedby the strictly proper transfer function

M(s) = km nm(s)dm(s)

= kms + am

,

input yr and output ym.

The purpose is to find a control law u(t), using onlythe plant input and output measurements, such that theoutput error

e0 = y ym (15)tends to zero asymptotically for arbitrary initial conditionsand arbitrary piece-wise continuous uniformly boundedreference signals yr(t).

Let us consideru = T (16)

where1 =

ap amkp

, 2 =km

kp(17)



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Model

M(s)

Plant

W(s)

Relay

Relay

2

1

2 = 2 sgn(e0 yr)

1 = 1 sgn(e0 y)

yr ym

yu

2

1

e0-

+

+

+

Figure 1. Block diagram of variable structure model refe-rence adaptive controller (VS-MRAC)

as the control signal to the plant (W(s)) matches themodel reference (M(s)) exactly, i.e., the transfer functionof the closed-loop plant, from yr to y is M(s). Of course,T can only be known ifW(s) is known. When this is notthe case the control signal is

u = T, (18)

where T = [1 2] is the vector of adaptive parameters(under some signal richness condition ) and

= [y yr]T

, (19)

is defined as regressor vector.

The following usual assumptions are made:

1. the relative degree n of the plant W(s) is knownand the model reference M(s) has the same relativedegree;

2. only the plant input and output are used to generatethe control signal u;

3. the order of the plant is known, say n, i.e., dp(s) ismonic of order n;

4. the plant and the model are supposed to be com-pletely observable and controllable (the pairs of monicpolynomials (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

i = i sgn(e0i)

wherei > |

i |, i = 1, 2

4. CONTROLLER STRUCTURE

This work proposes, using a modified Hirschorns inversesystem method, to decouple the nonlinear MIMO systemto get a number of SISO linear systems (see Figures 2 and3), in which each input affects only one output. The Figure2 presents a block diagram of Hirschorns inverse systemmethod. It is important to observe that a number of SISOlinear systems is , where m.

The modified Hirschorns inverse system method proposed

(see Figure 3) here is obtained by change

A(

x) (system

13). The main idea behind this modification is change the

linear decoupled transfer functions of s

toW(s) =

bwnsn + + bw1s + bw0

sn + awn1sn1 + + aw1s + aw0(20)

Left

InverseSystem

ur1

.

.

.ur

MIMO

System

u1

.

.

.um

y1

.

.

.yl

s

.

.

.

s

ur1

ur

y1

y

Figure 2. Block diagram of Hirschorns inverse systemmethod

with relative degree

Left

InverseSystem

ur1

.

.

.ur

MIMO

System

u1

.

.

.um

y1

.

.

.yl

W1(s)

.

.

.

Wm(s)

ur1

ur

y1

y

Figure 3. Block diagram of the modified Hirschorns in-verse system method

The modified Hirschorns inverse system method will de-couple the nonlinear MIMO system to get a number ofSISO linear systems with Wi(s) as linear decoupled trans-fer functions. But, when there is parameter uncertainty in

nonlinear MIMO system, the linear decoupled functionscan be interpreted as (see Figure 4)

yi = W xi(s)(uri + di) (21)

wheredi = f(u1, , um) (22)

is an input disturbance,

W xi(s) = Wi(s) + (23)

and is an unmodeled dynamic.

uriWxi(s)

yi

di

+

+

Figure 4. Linear decoupled functions when there is para-meter uncertainty in nonlinear MIMO system

To ensure that modified Hirschorns inverse system methodwill decouple the nonlinear MIMO system, we are using aVS-MRAC controller (Vi), to each decoupled linear system(see Figure 5). The VS-MRAC controller offers remarka-ble stability and performance robustness properties withrespect to parametric uncertainties, unmodelled dynamicsand external disturbances.

uriWxi(s)

yi

di

Viyri

+

+

Figure 5. VS-MRAC controller applied to the system ob-tained using the modified Hirschorns inverse systemmethod

The proposed controller decouples the MIMO linear sys-tem into that of a number of SISO linear systems withtransfer function given by the model reference M(s) ofVS-MRAC controller (see Figure 6).

5. SIMULATION RESULTS

This section presents an example which highlights theperformance of the proposed controller. Therefore, the



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yiMi(s)

yri

Figure 6. Set of decoupled linear transfer functions ob-tained using proposed controller

proposed controller is applied to a modified Chuas circuit,

which is very sensitive to variations in its components andhas a strong coupling between inputs and outputs. Themodified Chuas circuit in state-space is given by

y1 = k1(y2 y1) k2g(y1) + u1y2 = k3(y1 y2) + k4y3 + u2y3 = k5y2 + u2

(24)

where

g(x) = Gbx + 0.5(Ga Gb)[|x + Bp| |xBp|] (25)

is a nonlinear function and Ga, Gb, Bp, ki (i = 1, . . . , 5)are auxiliary constants depending on physical componentsof the circuit.

5.1 Decoupling the modified Chuas circuit

The aim is to decouple the nonlinear MIMO system (24)into two linear SISO systems (see Figure 7), one linearsystem to y1 and one to y2, where y1 and y2 represent thevoltage in two different capacitors of the circuit.

Left

InverseSystem

ur1

ur2

Modified

Chuas

Circuit

u1

u2

y1

y2

W1(s)

W2(s)

ur1

ur2

y1

y2

Figure 7. Block diagram of the modified Hirschorns in-verse system method

The left-inverse system for (24), using the method pre-sented in section 2, is given by

y1 = y1 + ur1y2 = y2 + ur2y3 = k5y2 (26)

withu1 = k1(y2 y1) + k2g(y1) + ur1u2 = k3(y1 y2) k4y3 + ur2 (27)

If we change (26) to

y1 = kmy1 + kmur1

y2 = km

y2 + kmur2

y3 = k5y2(28)

where km is defined by the model reference of VS-MRACcontroller, the modified Hirschorns inverse system is ob-tained.

5.2 Design of VS-MRAC controllers

First of all, it is necessary to define the reference models(M1(s), M2(s))

Mi(s) =ymi

yri=

km

s + km

The second step is to define the control law to eachdecoupled linear system

ur1 = Tv11

ur2 = Tv22

(29)

whereTi = [yi yri]

Tvi = [i 1 i 2](30)

with i = 1, 2. Thus, the parameter update law is

1 1 = 1 1 sgn(e0 1y1)

1 2 = 1 2 sgn(e0 1yr1)2 1 = 2 1 sgn(e0 2y2)2 2 = 2 2 sgn(e0 2yr2)

(31)

wheree0 i = yi ymi

5.3 Simulations

Consider the modified Chuas circuit (system 24) withinitial conditions

x(0)y(0)z(0)

=

0.152640.022810.38127

, (32)

and k1 = 7k2 = 10k3 = 0.35k4 = 0.5k5 = 7

Bp = 1Ga = 4Gb = 0.1

(33)

In the design of VS-MRAC controller, the model referenceis chosen km = 20 and

1 1 = 0.33751 2 = 1.42502 1 = 0.27562 2 = 0.3937

(34)

Another important consideration is that all simulationshas 400s and at t > 250s the parameters values of thesystem will change to

k1 = 10k2 = 12.5k3 = 0.363k4 = 0.454k5 = 8.139

(35)

in the simulations of Figures (9-11).

The chaotic behavior of modified Chuas circuit is shownin Figure 8. The Figures (e-f) presents the behavior ofsystem (24) when the inputs u

1and u

2has form given by

the Figures 8(c-d), respectively.

The simulation of the Figure 8 (a-b) shows that thebehavior of the modified Chuas circuit remains chaoticdespite input signals different from zero.

Next simulation (Figure 9) presents the behavior of system(24) using (28-27) as left-inverse system. In this simulationit is important to note that any constant value to ur2 leadsthe system to instability 1 .

The behavior of system using the proposed left-inversesystem (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 havedecoupled the system perfectly. That was not the case due torounding in the parameters and the sensitivity of the chaotic system.



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(a) y3 y1 (b) y1 y2

(c) t u1 (d) t u2

(e) t y1 (f) t y2

Figure 8. Simulation of modified Chuas circuit, withdifferent values to the inputs u1 and u2

(a) t (y1 and ur1) (b) t (y2 and ur2)

(c) t u1 (d) t u2

Figure 9. Simulation of modified Chuas circuit with left-inverse system

error increases. Despite inverse system, there is a coupling

between the inputs and outputs of the system. A strongcoupling between ur2 and y1, which makes the systemunstable, and a weak coupling between ur1 and y2.

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

(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

Figure 10. Simulation of modified Chuas circuit with left-inverse system and VS-MRAC controllers

In this simulation (Figure 10) the focus is the behaviorof the proposed controller, which has a fast transientand a small chattering 2 on the output signal. Anotheraspect is that proposed controller is robust to parametricuncertainties and input disturbances.

Finally, the Figure 11 shows the behavior of system (24)without left-inverse system and using the VS-MRAC con-trollers (section 5.2) with

1 1 = 12.831 2 = 54.152 1 = 6.892 2 = 9.84

(36)

which are the smaller values to VS-MRAC controller en-sure that the output error e0 tends to zero asymptotically.

According to (21), any adaptive controller with a fasttransient can be used to decouple a nonlinear MIMOsystem 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 existenceof non-ideal relays, having hysteresis effects and finite time delays inswitching of the output.



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(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

Figure 11. Simulation of modified Chuas circuit withoutleft-inverse system and using designed VS-MRAC

controllersthe proposed controller will improve system performanceby reducing the output chattering. The VS-MRACcontroller used was designed to have the smallest possibleoutput chattering.

Although the VS-MRAC controller used was designedto have the smallest possible output chattering, theproposed controller has a lower output chattering. Thissimulation shows that the application of the inverse systemsmoothes the control signal and thereby reduces the effectof chattering in the output.

6. CONCLUSION

In this work, a new control structure, based on the unionbetween VS-MRAC controllers and the technique of left-inverse system, was proposed. This structure uses onlyinput/output measurements, improves the transient per-formance (reducing the output chattering) and it isrobust to parametric uncertainties and disturbances. Allthese features are demonstrated by simulation results ofa simple electronic circuit that exhibits chaotic behavior(Chuas circuit). This structure may be used for SISO sys-tems, linear or nonlinear, to smooth the VS-MRAC controlsignal, thus reducing the chattering in the output signal.Finally, this paper presented an application in electronic

circuits, which can be expanded to other electronic circuitsas switched-mode power supply, modulators, digital-to-analog converter, etc.

In future works, the stability analysis, applications onindustrial environments will be discussed.

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