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Output mini-max control for polynomial systems:analysis and applicationsManuel Jiménez-Lizárragaa, Michael V. Basina, Victoria Celeste Rodríguez Carreóna & PabloCesar Rodríguez Ramíreza

a Facultad de Ciencias Físico-Matemáticas, Universidad Autónoma de Nuevo León, SanNicolás de los Garza N. L., MéxicoPublished online: 21 Jan 2013.

To cite this article: Manuel Jiménez-Lizárraga, Michael V. Basin, Victoria Celeste Rodríguez Carreón & Pablo Cesar RodríguezRamírez (2014) Output mini-max control for polynomial systems: analysis and applications, International Journal of SystemsScience, 45:9, 1880-1891, DOI: 10.1080/00207721.2012.757385

To link to this article: http://dx.doi.org/10.1080/00207721.2012.757385

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International Journal of Systems Science, 2014Vol. 45, No. 9, 1880–1891, http://dx.doi.org/10.1080/00207721.2012.757385

Output mini-max control for polynomial systems: analysis and applications

Manuel Jimenez-Lizarraga∗, Michael V. Basin, Victoria Celeste Rodrıguez Carreon and Pablo Cesar Rodrıguez Ramırez

Facultad de Ciencias Fısico-Matematicas, Universidad Autonoma de Nuevo Leon, San Nicolas de los Garza N. L., Mexico

(Received 4 January 2012; final version received 27 November 2012)

This paper presents a solution to a robust optimal regulation problem for a nonlinear polynomial system affected by parametricand matched uncertainties, which is based only on partial state information. The parameters describing the dynamics of thenonlinear polynomial plant depend on a vector of unknown parameters, which belongs to a finite parametric set, and theapplication of a certain control input is associated with the worst or least favourable value of the unknown parameter. Ahigh-order sliding mode state reconstructor is designed for the nonlinear plant in such a way that the previously designedcontrol can be applied for a system with incomplete information. Additionally, the matched uncertainty is also compensatedby means of the same output-based regulator. The obtained algorithm is applied to control an uncertain nonlinear inductorcircuit of the third order and a mechanical pendulum of the third order, successfully verifying the effectiveness of thedeveloped approach.

Keywords: mini-max control; nonlinear systems; sliding mode observers; electrical circuit

1. Introduction

Since the publication of the ground-breaking paper writtenby Kalman (1960), the foundation of what later becameknown as the linear-quadratic regulation problem was laiddown. This paper had a major impact in the control field andinfluenced on many control practitioners and researchers.Eventually, after this publication and explosion of interest,the need to extend the developed approach became evidentfor some more complex or general situations, such as non-linear dynamics, uncertainties in the model and incompleteinformation on the state variables. In many applications,the originally linear modelling cannot fit all the situationsin practice, which are mostly nonlinear by nature. For in-stance, the two main approaches to derive the optimal con-troller for nonlinear systems are the general principles of themaximum of Pontryagin (Kwakernaak and Sivan 1972) andthe dynamic programming (Bellman 1957). Unfortunately,those tools do not provide an explicit form for the opti-mal control in most cases. Nevertheless, there is actually along tradition of the optimal control design for nonlinearsystems (see, for example, Al’brekht 1961; Yoshida andLoparo 1989) and the optimal closed-form filter design fornonlinear plants (Wonham 1965; Benes 1981; Yau 1994;Yan, Lam, and Xie 2003), in particular, polynomial ones(Basin 2003; Basin, Calderon-Alvarez, and Skliar 2008).The second critical point is the existence of uncertaintiesin the model and the need to optimise the system (see Shiand Shue 1997; Guan, Shi, and Liu 2002; Basin, Shi, andCalderon-Alvarez 2009; Basin and Shi 2010; Basin, Shi,

∗Corresponding author. Email: manalejimenez@yahoo.com

and Soto 2010; Basin, Shi, and Calderon-Alvarez 2011;Shi, Agarwal, Boukas, and Shue 2011). Such a problemcan of course be viewed from different perspectives andapproaches: one of them is to consider that system parame-ters belong to an uncertain parametric finite set, where eachvalue of the parameters characterises one possible dynam-ics of the model, and the designed robust control shoulddeal with all of them simultaneously. For these kind of op-timisation problems, the robust version of the traditionalmaximum principle referred to as Robust Maximum Prin-ciple (RMP) (see Boltyanski and Poznyak 1999; Boltyanskiand Poznyak 2002; Poznyak, Duncan, Pasik-Duncan, andBoltyanski 2002; Jimenez-Lizarraga and Poznyak 2007) al-lows one to design an ‘optimal policy’ of a mini-max typefor a multi-parameter problem, where each set of param-eters is viewed as a possible parametric realisation of thedynamic equation. Another important class of uncertain-ties is the matched uncertainty, which has been extensivelystudied in the theory of variable structure systems and slid-ing mode control. Recent advances in high-order slidingmode control and observation have resulted in algorithmsto design a wide class of control laws based on the outputmeasurements. The capability of this approach is revealedby the development of practical arbitrary-order real-timerobust exact differentiators (Levant 1993, 1998). The High-Order Sliding Mode (HOSM) differentiators are used in ad-vanced HOSM-based observers for estimation of the systemstate in the presence of unknown external disturbances (seeBejarano, Fridman, and Poznyak 2007a; Fridman, Levant,

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International Journal of Systems Science 1881

and Davila 2007; Fridman, Shtessel, Edwards, and Yan2008; Wu, Shi, and Gao 2010; Juarez-Lopez, Camacho-Nieto, and Chairez 2012; Khan, Iqbal, Iqbal, and Ahmed2012), which are also reconstructed online. The feasibil-ity and applicability of the HOSM observers are con-vincingly demonstrated in practical situations (see Levant,Pridor, Gitizadeh, Yaesh, and Ben-Asher 2000; Shi, Xia,Liu, and Rees 2006; Benallegue, Mokhtari, and Fridman2008; Daniele, Capisani, Ferrara, and Pisu 2008; Fraguela,Fridman, and Alexandrov 2012a,b; Lin, Chang, and Hsu2012; Wu, Su, and Shi 2012).

In this paper, we face the problem of designing a regu-lator for a nonlinear polynomial system affected by uncer-tainties in the parameters (multi-parameter problem) anduncertainties of the matched type, when only output mea-surements are available (see Wang, Lam, Xu, and Gao 2005;Shu and Lam 2009; Basin, Shi, and Calderon-Alvarez 2010;Wang and Gao 2011). To the best of authors’ knowledge,this problem, involving incomplete state measurements, hasnever been studied for nonlinear polynomial systems (only amulti-parameter problem was studied; Jimenez-Lizarraga,Basin, and Rodriguez-Ramirez 2012) and is of great in-terest for practice due to a broad range of applications,as shown in the application section. We combine the afore-mentioned tools to effectively design the robust output mini-max controller for nonlinear polynomial systems (see alsoBasin, Rodriguez-Gonzalez, Fridman, and Acosta 2005;Basin, Ferreira, and Fridman 2007; Bejarano, Fridman, andPoznyak 2007b). Firstly, using the robust maximum princi-ple, a mini-max control is presented for a nonlinear poly-nomial system, then, a second Integral Sliding Mode (ISM)control is designed to deal with the matched uncertainty.These two control laws are based on the state reconstruc-tion by means of a HOSM observer for the nonlinear plant,which rapidly recovers the state signals in finite time. Notethat applying the RMP technique we are able to deal withparametric uncertainties. However, the main drawback ofthis approach is that we have to handle a set of uncer-tainties, which are not a priori recognisable as matched orunmatched. Therefore, in case of unmeasured but matcheduncertainties, another approach should be used. Remark-ably, the ISM controller, employed in this paper, provides areasonable and feasible solution. To the best of the authors’knowledge, neither of the previously mentioned papers, norother works, attempted to solve the output mini-max controlproblem for polynomial systems, considering unmatcheddisturbances (parameters), matched uncertainties as well asincomplete state information.

As an illustration of the developed algorithm, two appli-cation examples are presented. The first one is the Duffingequation, which is used to model a circuit including a non-linear inductor of the third order, where the parametersdescribing the dynamics of the circuit belong to a two-parametric set and only one of the states is measurable. Thecontrol problem for such a circuit has attracted consider-

able attention of the control community, since it deals witha complex chaotic nonlinear system, which has a varietyof applications ranging from physics to engineering (seeKapitaniak 1996; Loria, Panteley, and Nijmeijer 1998).Given the complex behaviour of the circuit, the control taskbecomes really challenging. Our approach demonstratesthat this circuit can be effectively controlled in the presenceof uncertainties and incomplete state information. The sec-ond example deals with a control problem for a pendulum,whose dynamics is approximated using the Taylor expan-sion up to the third order around the equilibrium point,instead being of linearised. This allows us to model thependulum dynamics more precisely. The control problemis two-parametric, only the pendulum angle can be mea-sured, and a matched perturbation affecting the velocity isconsidered. Again, our approach shows good performancedealing with uncertainties and partial information.

The rest of the article is organised as follows. Section 2presents the model description, problem statement and basicassumptions. Section 3 yields the mini-max control solu-tion for a polynomial system. In Section 4, the HOSM stateobservers are developed for the stated problem. Section 5designs the output ISM control. In Section 6, the designedalgorithm is applied to an electrical circuit including a non-linear inductor of the third order and a pendulum system ofthe third order. Section 7 concludes this study.

Remark 1: The time-dependence notation is intentionallyomitted in some places where it is clear from the previouscontext.

2. Problem statement

Consider the following nonlinear differential equation witha polynomial term including an unknown vector α, whichintroduces the uncertainty into the model, and an outputequation as the available state information:

x (t ; α) = f (t, x, α) + B (t ; α) u (t)+ d (t ; α) + B (t ;α) ζ (t) ,

y (t ; α) = C (t ; α) x (t ; α) x (t0; α) = x0,

(1)

where x (t ; α) ∈ Rn is the state vector, u(t) is the con-

trol input that varies within a given control region U ⊂R

m, d(t ; α) ∈ Rn is considered a sufficiently smooth (up

to the nth time derivative) function for each fixed param-eter α, B(t ; α) ∈ R

m×n is the control known matrix, andα is a parameter which belongs to a given parametric setA, ζ (t) ∈ R

nis an uncertainty of the matched type whichbelongs to the span of the control matrix and is assumedto be bounded by a known scalar function q (t), that is,‖ζ (t)‖ ≤ q (t), C (t ; α) ∈ R

p×n is the output matrix, andy (t ; α) ∈ R

p the output signal. In this paper, we considerA as a finite set, that is, A = {α1, α2, . . . , αN }, each onerepresenting a possible realisation or possible model of thesystem. The time variable runs in an interval t ∈ [t0, T ].

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1882 M. Jimenez-Lizarraga et al.

Note that in this paper we face two types of uncertainty,so the control challenge is to design a law that deals effec-tively with both and is based only on the output information.For the unmatched type the solution or ‘optimality’ can begiven in terms of the worst case of the performance index,and for the matched uncertainty we propose the ISM basedon the output. The control is designed in two steps, that is,there are two terms in the control law: u = u1 + u2. Thefirst part of the control is intended to deal with the paramet-ric uncertainty and the second part to deal with the matchedpart of the uncertainty.

For each fixed parameter α, the system (1) is assumedto be uniformly controllable and observable for everyt ∈ [t0, T ]; the definitions of the uniform controllability andobservability for nonlinear systems can be found in Isidori(2001). We assume also that dim (u) = m ≤ n = dim (x)and dim (y) = p ≤ n = dim (x). For each fixed parameterα, the nonlinear function f (t, x, α) is considered as a poly-nomial of n variables, which are the components of the statevector x (t ; α) ∈ R

n; this requires a special definition of thepolynomial for degrees n > 1. Following the previous work(see Basin, Perez, and Skliar 2006), a p-degree polynomialof a vector x (t ; α) ∈ R

n is regarded as a p-linear form of n

components of x (t), that is

f (t, x, α) = a0 (t ; α) + a1 (t ; α) x + a2 (t ; α) xxᵀ

+ · · · + as (t ; α) x · · · s times · · · x. (2)

Here, the involved parameters are: a0 is a vector of di-mension n, a1 is a matrix of dimension n × n, a2 is a3D tensor of dimension n × n × n, and as is an (s + 1)Dtensor of dimension n × · · · (s + 1) t imes · · · × n, andx × · · · s times · · · × x is a pD tensor of dimension n ×· · · s times · · · × n, obtained by p times spatial multiplica-tion of the vector x by itself. It is also possible to representsuch a polynomial in a summation form

fk (t, x, α)

= a0 k (t ; α) +∑

i

a1 ki (t ; α) xi +∑ij

a2 kij (t ; α) xixj

+ · · · +∑i1...is

a1 ki1...is (t ; α) xi1 . . .xis ,

k, i, j, i1. . .is = 1, . . ., n, (3)

where the dependence of the a0 (t ; α), a1 (t ; α) , . . . , as(t ;α) on α means that a tensor of parameters also belongs tothe parametric set.

Remark 1: Clearly, the uncertainty in the realised param-eters is represented by a value of α. The parameter α be-longs to a finite setA that contains all the possible scenariosor parametric realisations of the nonlinear plant, which isfixed during the actual process with no possibility of changeonce the process has started. So, each trajectory is uniquely

determined by a given set of parameters. Nevertheless, thereis no information on which trajectory is realised. In this way,the proposed control should deal with all the parameters andprovide acceptable behaviour for such a class of systems.

The following quadratic cost functional is defined:

g (x (t ; α) ,u1 (t) ,α)

= 1

2xᵀ (T ; α) Qx (T ; α) + 1

2

∫ T

t1

(xᵀ (t ; α) Lx (t ; α)

+uᵀ1 (t) Ru1 (t)) dt. (4)

This performance index is given in the standard Bolza form,where R is a strictly positive definite symmetric matrix, L

and Q are two non-negative definite symmetric matrices.The design of the control is carried out in two steps, first,for the multi-parameter problem the solution is formulatedas an extremum of the following mini-max problem:

minu(t)

maxα∈A

g (x (t ; α) , u1, α) . (5)

Then, at the second step, an integral sliding mode is de-signed to eliminate the matched uncertainty from almostthe beginning of the process. Both controls are based on anHOSM observer.

3. Mini-max control for polynomial system

Let us proceed first with designing the mini-max control.Consider the following extended system including all of thescenarios without the matched uncertainty:

x (t) = f (t, x (t)) + B (t) u1 (t) + d (t) , (6)

f :=

⎡⎢⎣f (t, x, α1) · · · 0... .

...0 · · · f (t, x, αN )

⎤⎥⎦; x :=

⎛⎜⎝x(t, α1)...

x(t, αN )

⎞⎟⎠ai :=

⎡⎢⎣ai .. 0...

. . ....

0 .. ai

⎤⎥⎦; Bᵀ :=

⎡⎢⎣ Bᵀ (t, α1)...

Bᵀ (t, αN )

⎤⎥⎦, (7)

i = 0, . . ., s

where the tensor terms also appear in an extended form forthe coefficients of the polynomial. The extended vector dis defined as

dᵀ := (dᵀ (t ; α1) , . . . , dᵀ (t ; αN )) . (8)

Note that the dependence on the uncertain parameter α

has disappeared; the new dynamics includes the completefamily of plants but the control remains the same for allplants. The mini-max regulator that realises (5) with respect

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to the quadratic criterion (4) for the polynomial system (1)according to Jimenez-Lizarraga et al. (2012) takes the form

u1 = −R−1Bᵀ [Mλx + pλ] , (9)

where the matrix function Mλ is the solution of the follow-ing Riccati-like equation:

Mλ + �L + [a1 + 2a2x + 3a3xxᵀ + · · ·+ sasx· · · (s − 1) t imes · · ·x]ᵀMλ + Mλ[a1 + a2x

+ a3xxᵀ + · · · + asx· · · (s − 1) t imes · · ·x]

− MλBR−1BᵀMλ = 0 (10)

with terminal condition Mλ (T ) = �Q, and the parame-terised vector function pλ is solution of the linear equation

pλ + Mλa0 + [a1 + 2a2x + 3a3xxᵀ + · · ·+ sasx· · · (s − 1) t imes · · ·x]ᵀpλ − MλBR−1Bᵀpλ

+ Mλd = 0 (11)

with terminal condition as pλ (T ) = 0.The matrix � con-taining the optimal weight belongs to the followingsimplex:

SN :={

λ ∈ RN : λi ≥ 0;

N∑i=1

λi = 1

}. (12)

The matrices L, Q, � are defined as

L :=

⎡⎢⎣L · · · 0... .

...0 · · · L

⎤⎥⎦; Q :=

⎡⎢⎣Q · · · 0... .

...0 · · · Q

⎤⎥⎦� :=

⎡⎢⎣λ1In×n .. 0... .

...0 . λ2In×n

⎤⎥⎦ . (13)

The matrix � contains the optimal weight parameters λ =λ∗ that solve the following optimisation problem:

λ∗ = minλ∈SN

J (λ) ,

J (λ) := maxα∈A

g (x (t ; α) , u1, α) ,(14)

with u1(t) given in Equation (9), which is parameterised bythe vector λ = (λ∗

1, λ∗2, . . ., λ

∗N )ᵀ (λ∗

i ∈ SN ) through (10)and (11). Note that to obtain the optimal weight vectorsolving the problem may not be an easy task. We followJimenez-Lizarraga et al. (2012) to provide a feasible nu-merical procedure to find the mini-max weights in the caseof two scenarios. Assuming that there exists an estimation

process for the states of the extended system x, the mini-max control law is implemented as

u1 = −R−1Bᵀ [Mλx + pλ] . (15)

4. HOSM observer design for polynomial system

Consider again the extended system with partial state mea-surements and without matched uncertainty

x (t) = f (t, x (t)) + B (t) u1 (t) + d (t) ,

y (t) = Cx (t) ,(16)

where

C :=

⎛⎜⎜⎝C(t, α1) · · · 0

.... . .

...

0 · · · C(t, αN )

⎞⎟⎟⎠ . (17)

As mentioned, the HOSM observer is based upon the exactdifferentiator of Levant (1998), following Fridman et al.(2008). The observer takes the form

·xi,1 = wi,1 = −κri+1M

1/(ri+1)i |yi − yi |(ri )/(ri+1)

× sign(yi − yi) + xi,2·xi,2 = wi,2 = −κri

M1/(ri )i |xi,2 − wi,1|(ri−1)/(ri )

× sign(xi,2 − wi,1) + xi,3...·

xi,ri= wi,ri

= −κ2M1/(2)i |xi,ri

− wi,ri−1|(1)/(2)

× sign(xi,ri− wi,ri−1) + xi,ri+1·

xi,ri+1 = −κ1Misign(xi,ri+1 − wi,ri)

yi = xi,1

(18)

where ri is the relative degree with respect to the controlsignal of the i component of the output y(t) = Cx(t); theprocedure is repeated for each element of the output. Thesolutions of these equations yielding the state estimates are

xTi = [

xTi,1 xT

i,2 . . . xTi,ri

].

Now, define as x the set of solutions for each componentof y(t).

xT = [xT

1 xT2 . . . xT

m

].

It is shown (Davila, Basin, and Fridman 2010) that theobserver error converges to zero for a finite time if theparameters M and κi are sufficiently large. Note thatthe observer (18) has a recursive form, which is useful forthe parameter adjustment. The specific parameter valuesare given in the first example (Section 6).

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1884 M. Jimenez-Lizarraga et al.

5. Output integral sliding mode for uncertainpolynomial system

In this section, we present the design of the control u2

to deal with the matched part of an uncertainty, which isbased on the output integral sliding mode. Consider againthe extended system

x (t) = f (t, x (t)) + B (t) (u1 (t) + u2 + ζ (t)) + d (t) ,

y (t) = Cx (t) . (19)

Note that the matched part of an uncertainty is the samefor all the models. Our purpose is to eliminate such anuncertainty as fast as possible, so that the mini-max controldoes the rest of the job. To design the output ISM, definethe following auxiliary output sliding surface:

z(y(t)) := Gy(t) − σ (t), (20)

where G = (CB)+ = [(CB)T (CB)]−1(CB)T . Evaluatingthe time derivative of z(y(t)) yields

z(y(t)) = GC(f(x, t) + B(u1(t) + u2(t) + ζ (t))

+ d(t)) − σ (t). (21)

Since we do not have the complete state information, let usdefine σ (t) based only on the estimate x

σ (t) = GC(f (x, t) + Bu0(t) + d(t))

σ (0) = Gy(0), (22)

which leads to

z(y(t)) = GC(f(x, t) − f (x, t)) + u1(t) + ζ (t) (23)

z(0) = 0.

The second part of the control u2, which is discontinuous,is proposed as

u2 = −γ (t)z(t)

‖z(t)‖ (24)

with the scalar γ (t) satisfying

γ (t) − q(t) − ‖GC‖‖f(x, t) − f (x, t)‖ ≥ ρ ≥ 0,

ρ is a constant.To correctly choose the parameters of the discontinuous

control law and ensure the movement of the system alongthe sliding surface, let us propose the Lyapunov function asV = (1/2)‖z‖2. The time derivative is calculated as

V = zT (GC(f(x, t) − f (x, t)) − γ (t)z(t)

‖z(t)‖ + ζ (t))

≤ −‖s‖(γ (t) − q(t) − ‖GC‖‖f(x, t) − f (x, t)‖)

≤ −‖s‖ρ ≤ 0.

Because of the selection of the initial condition for z (23),z(0) = 0, we obtain

V (s(t)) ≤ V (s(0)) = 1

2‖s(0)‖2 = 0,

which implies that s(t) = s(t) = 0. The conclusion is thatthe ISM does not have the reaching phase.

Therefore, the equivalent control is obtained as

ueq = −GC(f(x, t) − f (x, t)) − γ (t). (25)

Substituting the equivalent control (25) into Equation(19) yields

x = f(x, t) − BGC(f(x, t) − f (x, t)) + Bu0(t) + d(t)

y(t) = Cx(t), (26)

which is the extended system, where the matched part ofthe uncertainty is eliminated and only a term of dependingon the observation appears; such a term vanishes as theobserver converges to the real states.

Remark 1: Given that the ISM eliminates the matched partof the uncertainty, the combination of the two techniques,the mini-max control and the ISM along with the HOSMfast state reconstruction, presents a powerful tool for con-trolling polynomial systems with partial state information,as illustrated in the following numerical examples.

6. Application examples

6.1. Nonlinear circuit

The well-known nonlinear electric oscillator including anonlinear inductor (see Figure 1) is modelled by a bi-dimensional polynomial system of the third order (seeNijmeijer and Harry 1995; Loria et al. 1998). Considerthe case of two possible scenarios (N = 2) of the followingcircuit:

x1(t, 1) = x2 (27)

x2(t, 1) = −1.1x1 − 0.4x2 − x31 + 2.05 cos(1.8t)

+ sin (1000t) − u (28)

Figure 1. Nonlinear circuit.

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x1(t, 2) = x2

x2(t, 2) = −1.15x1 − 0.45x2 − 1.05x31 + 2 cos(1.9t)

+ sin (1000t) − u, (29)

where the known uncertainty is d (t, 1) = 2.05 cos(1.8t),and the unknown matched uncertainty is ζ (t, 1) =sin (1000t) , the relation to Figure 1 is givenby α1/C = −1.1, 1/RC = −0.4, E0/R = 2.05 , α2/C =−1, ω = 1.8, withα1 and α2 being operation constants, andthe following output measurements for the two models:

y(t, 1) = x1(t, 1)

y(t, 2) = x1(t, 2).

The equations for the state reconstructor are given by

·x1,1 = w1 = x2 − κ3M

1/3|y1 − y1|2/3sign(y1 − y1)·x1,2 = w2 = x3 − κ2M

1/2 |x2 − w1|1/2sign(x2 − w1)·x1,3 = κ1M sign(x3 − w2)

y1 = x1,

·x2,1 = w4 = x5 − κ6M

1/3|y2 − y2|2/3sign(y2 − y2)·x2,2 = w5 = x6 − κ5M

1/2 |x5 − w4|1/2sign(x5 − w4)·x2,3 = κ4Msign(x6 − w5)

y2 = x4,

where x1,1, x1,2, x2,1, x2,2 are the estimates of the state vari-ables x1(t, 1), x2(t, 1), x1(t, 2) and x2(t, 2), respectively.The initial conditions for the observer variables are set tozero. The following constants are used for tuning up the

observer:

M = 1, κ1 = κ2 = κ4 = κ5 = 20, κ3 = κ6 = 10.

Finally, the control law takes the form

u1 = (P21 + P41)x1 + (P22 + P42)x2 + (P23 + P43)x3

+ (P24 + P44)x4 + p2 + p4.

The figures are organised as follows. Figures 2 and 3show the state variables and estimates of the models 1 and2. Note that the states of both plants demonstrate identicalbehaviour, when the robust control is applied, thus provingthat the robust design works well for any of the two plants.The estimates also look almost the same all the time be-cause of the fast convergence rate of the observer. Figure 4presents the values of the cost depending on λ, where wecan see that the performance index has the minimum (ofthe worst case value of the cost) around λ = 0.28. Figure 5corresponds to the mini-max control based on the output.Finally, Figure 6 shows the ISM control based on the output.Again, we observe that with the application of the output-based ISM the perturbation does not affect the dynamics ofany of the plants, resulting in a good performance of boththe mini-max control and the ISM.

6.2. Pendulum

Consider the following equations for the simple pendulum(see Figure 7, where x = θ,m = 1):

d2x

dt2= −g

lsin(x) + u. (30)

Figure 2. States and estimates. Plant 1.

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Figure 3. States and estimates. Plant 2.

Figure 4. Performance index J (λ) .

The nonlinear term can be approximated by a polynomialusing the Taylor expansion of the function sin(x)

sin(x) = x − 1

6x3 + · · · + (−1)n+1x2n−1

(2n − 1)!+ · · ·

We consider the approximation by the third order poly-nomial, that is, Equation (30) is approximated as

d2x

dt2= −g

l

(x − 1

6x3

)+ u.

Proceeding with the change of variables x1 = x and x2 = x,

we obtain

x1 = x2

x2 = −g

l

(x1 − 1

6x3

1

)+ u.

The two-scenario model system becomes

x1(t, 1) = x2(t, 1)

x2(t, 1) = −3 ∗(

x1(t, 1) − 1

6x3

1 (t, 1)

)+ u + sen(1000t)

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Figure 5. Min-max control u1.

Figure 6. ISM control u2.

x1(t, 2) = x2(t, 2)

x2(t, 2) = −2.95 ∗(x1(t, 2) − 1

6x3

1 (t, 2)

)+u+sen(1000t),

with the output equations

y(t, 1) = x1(t, 1)

y(t, 2) = x1(t, 2),

and the performance index

g (x (t ; α) , u, α) = x21 (T , α) + x2

2 (T , α) +∫ T

0

(x2

1 (t, α)

+ x22 (tv) + u2(t)

)dt.

Figure 7. Simple pendulum.

Finally, the observer equations are·x1,1 = w1 = x2 − κ3M

1/3|y1 − y1|2/3sign(y1 − y1)

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·x1,2 = w2 = x3 − κ2M

1/2 |x2 − w1|1/2sign(x2 − w1)·x1,3 = κ1Msign(x3 − w2)

y1 = x1

·x2,1 = w4 = x5 − κ6M

1/3|y2 − y2|2/3sign(y2 − y2)·x2,2 = w5 = x6 − κ5M

1/2 |x5 − w4|1/2sign(x5 − w4)·x2,3 = κ4Msign(x6 − w5)

y2 = x2,

where the initial conditions for the observer variables areset to zero. The control law takes the form

u = (P21 + P41)x1 + (P22 + P42)x2 + (P23 + P43)x3

+ (P24 + P44)x4.

In this example, the figures are organised as follows:Figures 8 and 9 show the state variables and estimates of themodels 1 and 2. As in the preceding example, both plantsdemonstrate the same behaviour, when the robust controlis applied, thus proving that the robust design works wellfor any of the two plants. The estimates also look almostthe same all the time. For this case, the optimal value isλ = 0.8779. Figure 10 shows the estimation errors. Finally,Figure 11 shows the ISM control based on the output. Again,we observe that with the application of the output-based

Figure 8. States and estimates. Pendulum 1.

Figure 9. States and estimates. Pendulum 2.

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Figure 10. Estimation errors.

Figure 11. ISM control u2.

ISM the perturbation does not affect the dynamics of anyof the plants, resulting in a good performance through thecombination of the two control laws.

7. Conclusions

This paper presented a solution to the robust optimal regula-tion problem for an uncertain nonlinear polynomial systembased on output information. It was demonstrated that fora polynomial system with the parameters describing thedynamics of the nonlinear polynomial plant depending ona vector of unknown parameters, it is possible to design amini-max control law based on the output. A high-ordersliding mode observer was designed for the nonlinear plant

in such a way that the worst case or mini-max control can beapplied to a system with incomplete information. Addition-ally, a matched type uncertainty was compensated by meansof an output ISM control. The obtained algorithm was ap-plied to control an uncertain nonlinear electrical circuit ofthe third order and a mechanical pendulum of the thirdorder, successfully verifying the effectiveness of the devel-oped approach. The developed technique could be appliedto other technical nonlinear systems approximated by poly-nomial functions, such as an articulated robotic arm; an-other extension might involve unmatched uncertainties thatcannot be modelled in the framework of a multi-parameterproblem.

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AcknowledgementsM. Jimenez-Lizarraga was supported by project PAICyT CE851-11 and by the National Council of Science and Technology ofMexico (CONACYT) under project 169734. M. Basin was sup-ported by the National Council of Science and Technology ofMexico (CONACYT) under project 129081.

Notes on contributorsManuel Jimenez-Lizarraga received theBS degree in electrical engineering fromInstituto Tecnologico de Culiacan, Mexico,the MS and Ph.D. degrees on AutomaticControl from CINVESTAV-IPN Mexico,in 1996, 2000 and 2006 respectively. Heis currently with the Faculty of Physicaland Mathematical Sciences of AutonomousUniversity of Nuevo Leon, Mexico. His re-

search interests include uncertain differential games, robust andoptimal control and sliding mode observers.

Michael V. Basin received his Ph.D. de-gree in Physical and Mathematical Scienceswith major in Automatic Control and Sys-tem Analysis from the Moscow AviationInstitute in 1992. His work experience in-cludes Senior Scientist position in the Insti-tute of Control Science (Russian Academyof Sciences) in 1992–1996, Visiting Profes-sor position in the University of Nevada at

Reno in 1996–1997 and Full-Time Tenured Professor position inthe Autonomous University of Nuevo Leon, Mexico, from 1998.Dr. Basin published a monograph, more than 100 research pa-pers in international referred journals and more than 130 papersin Proceedings of the leading IEEE and IFAC conferences andsymposiums. He is the author of the ‘New Trends in OptimalFiltering and Control for Polynomial and Time-Delay Systems’,published by Springer. His works are cited more than 950 times.He is the Editor-in-Chief of Journal of The Franklin Institute, anAssociate Editor of Automatica, International Journal of SystemsScience, IET Control Theory and Applications, and other journals.Dr. Basin serves as a member of IEEE Control System SocietyTechnical Committee on Intelligent Control, Program Commit-tee member of IEEE Conference on Decision and Control 2008,IEEE Conferences on Control Applications 2009, 2012, IEEEWorkshops on Variable Structure Systems 2010, 2012, 2014. Dr.Basin was awarded a title of Highly Cited Researcher by ThomsonReuters (International Science Institute), the publisher of ScienceCitation Index, in 2009; he is a Senior Member of the IEEEControl Systems Society and a regular member of the MexicanAcademy of Sciences. His research interests include optimal fil-tering and control problems, stochastic systems, time-delay sys-tems, identification, sliding mode control and variable structuresystems.

Victoria Celeste Rodrıguez Carreon re-ceived her degree in mathematics in 2008,and at present is pursuing a Ph.D. in in-dustrial physics engineering at the Facultyof Physics and Mathematics Science fromthe Autonomous University of Nuevo Leon,Mexico. Her research interests include dif-ferential games, stochastic systems and con-trol problems.

Pablo Cesar Rodrıguez Ramırez receivedhis degree in mathematics at the Facultyof Physics and Mathematics Science fromthe Autonomous University of Nuevo Leon,Mexico, in 2008, is currently pursuing aPh.D. in industrial physics engineering inthe same unit. His research interests in-clude optimal filtering and control prob-lems, stochastic systems, identification and

sliding mode control, under the tutelage of Dr. Michael V. Basin.

ReferencesAl’brekht, E. (1961), ‘On the Optimal Stabilization of Nonlinear

Systems’, Journal of Applied Mathematics and Mechanics,25, 1254–1266.

Basin, M. (2003), ‘On Optimal Filtering for Polynomial SystemStates’, Transactions of the ASME. Journal of Dynamic Sys-tems, Measurement and Control, 125, 123–125.

Basin, M., Calderon-Alvarez, D., and Skliar, M. (2008), ‘OptimalFiltering for Incompletely Measured Polynomial States OverLinear Observations’, International Journal of Adaptive Con-trol and Signal Processing, 22, 482–494.

Basin, M., Ferreira, A., and Fridman, L. (2007), ‘Sliding ModeIdentification and Control for Linear Uncertain StochasticSystems’, International Journal of System Science, 38, 861–869.

Basin, M., Perez, J., and Skliar, M. (2006), ‘Optimal Filteringfor Polynomial System States With Polynomial Multiplica-tive Noise’, International Journal of Robust and NonlinearControl, 16, 287–298.

Basin, M., Rodriguez-Gonzalez, J., Fridman, L., and Acosta, P.(2005), ‘Integral Sliding Mode Design for Robust Filteringand Control of Linear Stochastic Time-delay Systems’, Inter-national Journal of Robust Nonlinear Control, 15, 407–421.

Basin, M., and Shi, P. (2010), ‘Central Suboptimal H∞ Filtering forNonlinear Polynomial Systems With Multiplicative Noise’,Journal of the Franklin Institute, 347(9), 1740–1754.

Basin, M., Shi, P., and Calderon-Alvarez, D. (2009), ‘Central Sub-optimal H∞ Filter Design for Nonlinear Polynomial Systems’,International Journal of Adaptive Control and Signal Process-ing, 23(10), 926–939.

Basin, M., Shi, P., and Calderon-Alvarez, D. (2010), ‘Approxi-mate Finite-dimensional Filtering for Polynomial States OverPolynomial Observations’, International Journal of Control,83(4), 724–730.

Basin, M., Shi, P., and Calderon-Alvarez, D. (2011), ‘Central Sub-optimal H∞ Control Design for Nonlinear Polynomial Sys-tems’, International Journal of System Sciences, 42(5), 801–808.

Basin, M., Shi, P., and Soto, P. (2010), ‘Mean-square Data-basedController for Nonlinear Polynomial Systems With Multi-plicative Noise’, Information Sciences, 195, 256–265.

Bejarano, F., Fridman, L., and Poznyak, A. (2007a), ‘Output Inte-gral Sliding Mode Control Based on Algebraic HierarchicalObserver’, International Journal of Control, 80, 443–453.

Bejarano, J., Fridman, L., and Poznyak, A. (2007b), ‘HierarchicalSecond-order Sliding-mode Observer for Linear Time Invari-ant Systems With Unknown Inputs’, International Journal ofSystems Science, 38, 793–802.

Bellman, R. (1957), Dynamic Programming, Princeton, NJ:Princeton University Press.

Benallegue, A., Mokhtari, A., and Fridman, L. (2008), ‘High-orderSliding-mode Observer for a Quadrotor Uav’, InternationalJournal of Robust and Nonlinear Control, 18, 427–440.

Dow

nloa

ded

by [

Uni

vers

ity o

f T

enne

ssee

At M

artin

] at

09:

35 0

4 O

ctob

er 2

014

International Journal of Systems Science 1891

Benes, V. (1981), ‘Exact Finite-dimensional Filters for CertainDiffusions With Nonlinear Drift’, Stochastics, 5, 65–92.

Boltyanski, V., and Poznyak, A. (1999), ‘Robust Maximum Prin-ciple in Minimax Control’, International Journal of Control,72(4), 305–314.

Boltyanski, V., and Poznyak, A. (2002), ‘Linear Multi-model TimeOptimization’, Optimal Control Applications and Methods,23, 141–161.

Daniele, B., Capisani, M., Ferrara, A., and Pisu, P. (2008), ‘FaultDetection for Robot Manipulators via Second-order Slid-ing Mode’, IEEE Transactions on Industrial Electronics, 55,3954–3963.

Davila, J., Basin, M., and Fridman, L. (2010), ‘Finite-time Param-eter Identification via High-order Sliding Mode Observer’, inProceedings of the American Control Conference, Baltimore,MD, USA, 2960–2964.

Fraguela, L., Fridman, L., and Alexandrov, V. (2012a), ‘Output In-tegral Sliding Mode Control to Stabilize Position of a StewartPlatform’, Journal of the Franklin Institute, Engineering andApplied Mathematics, 349, 1526–1542.

Fraguela, L., Fridman, L., and Alexandrov, V. (2012b), ‘Posi-tion Stabilization of a Stewart Platform: High-order SlidingMode Observers Based Approach’, Journal of the FranklinInstitute, Engineering and Applied Mathematics, 349, 441–455.

Fridman, L., Levant, A., and Davila, J. (2007), ‘Observation ofLinear System With Unknown Inputs via High-order Sliding-mode’, International Journal of System Science, 38, 773–791.

Fridman, L., Shtessel, Y., Edwards, C., and Yan, X. (2008), ‘High-order Sliding-Mode Observer for State Estimation and InputReconstruction in Nonlinear Systems’, International Journalof Robust and Nonlinear Control, 18, 399–413.

Guan, X., Shi, P., and Liu, Y. (2002), ‘Robust Output Feed-back Control for Uncertain Discrete Time Delay Sys-tems’, Systems Analysis Modelling Simulation, 42, 1829–1840.

Isidori, A. (2001), Nonlinear Systems, Berlin: Springer.Jimenez-Lizarraga, M., Basin, M., and Rodriguez-Ramirez, P.

(2012), ‘Robust Mini-max Regulator for Uncertain Non-linear Polynomial Systems’, IET Control Theory and Appli-cation, 6, 963–970.

Jimenez-Lizarraga, M., and Poznyak, A. (2007), ‘Robust NashEquilibrium in Multi-Model LQ Differential Games: Analysisand Extraproximal Numerical Procedure’, Optimal ControlApplications and Methods, 8(2), 117–141.

Juarez-Lopez, S., Camacho-Nieto, O., and Chairez, I. (2012),‘Non-parametric Modeling of Uncertain Hyperbolic Par-tial Differential Equations Using Pseudo-High Order SlidingMode Observers’, International Journal of Innovative Com-puting, Information and Control, 8, 1501–1521.

Kalman, R. (1960), ‘Contributions to the Theory of Optimal Con-trol’, Boletin de la Sociedad Matematica Mexicana, 5, 102–119.

Kapitaniak, T. (1996), Controlling Chaos, New York: Academic.Khan, Q., Iqbal, A., Iqbal, M., and Ahmed, Q. (2012), ‘Dynamic

Integral Sliding Mode Control for Siso Uncertain NonlinearSystems’, International of Innovative Computing, Informa-tion and Control, 8, 4621–4633.

Kwakernaak, H., and Sivan, R. (1972), Linear Optimal ControlSystems, New York: Wiley-Interscience.

Levant, A. (1993), ‘Sliding Order and Sliding Accuracy in Slid-ing Mode Control’, International Journal of Control, 58(6),1247–1263.

Levant, A. (1998), ‘Robust Exact Differentiation via Sliding ModeTechnique’, Automatica, 34, 379–384.

Levant, A., Pridor, A., Gitizadeh, R., Yaesh, I., and Ben-Asher,J. (2000), ‘Aircraft Pitch Control via Second Order SlidingMode’, Journal of Guidance, Control and Dynamics, 23,586–594.

Lin, T., Chang, S., and Hsu, C. (2012), ‘Robust Adaptive FuzzySliding Mode Control for a Class of Uncertain Discrete-time Nonlinear Systems’, International Journal of InnovativeComputing Information and Control, 8, 347–359.

Loria, A., Panteley, E., and Nijmeijer, H. (1998), ‘Control of theChaotic Duffing Equation With Uncertainty in all Parame-ters’, IEEE Transactions on Circuits and Systems-I Funda-mental Theory and Applications, 45, 1252–1255.

Nijmeijer, H., and Harry, B. (1995), ‘On Lyapunov Control ofthe Duffing Equation’, IEEE Transactions on Circuits andSystems-I Fundamental Theory and Applications, 42, 473–477.

Poznyak, A., Duncan, T., Pasik-Duncan, B., and Boltyanski, V.(2002), ‘Robust Maximum Principle for Multi-model LQ-problem’, International Journal of Control, 75(15), 1770–1777.

Shi, P., Agarwal, R., Boukas, E., and Shue, S. (2011), ‘Robust H∞

State Feedback Control of Discrete Time-delay Linear Sys-tems With Norm-bounded Uncertainty’, International Jour-nal of System Science, 42, 409–415.

Shi, P., and Shue, S. (1997), ‘H∞ Control of Interconnected Non-linear Sampled-data Systems With Parametric Uncertainty’,International Journal of System Science, 28, 961–976.

Shi, P., Xia, Y., Liu, G., and Rees, D. (2006), ‘On Designing ofSliding Mode Control for Stochastic Jump Systems’, IEEETransactions on Automatic Control, 51, 97–103.

Shu, Z., and Lam, J. (2009), ‘An Augmented System Approach toStatic Output-Feedback Stabilization With H∞ Performancefor Continuous-time Plants’, International Journal of Robustand Nonlinear Control, 19, 768–785.

Wang, Z., and Gao, H. (2011), ‘Distributed H∞ Filtering for Re-peated Scalar Nonlinear Systems With Random Packet Lossesin Sensor Network’, International Journal of System Science,42, 1507–1519.

Wang, Q., Lam, J., Xu, S., and Gao, H. (2005), ‘Delay-dependentand Delay-independent Energy-to-peak Model Approxima-tion for Systems With Time-varying Delay’, InternationalJournal of System Science, 36, 445–460.

Wonham, W. (1965), ‘Some Applications of Stochastic Differ-ential Equations to Nonlinear Filtering’, SIAM Journal onControl, 2, 347–369.

Wu, L., Shi, P., and Gao, H. (2010), ‘State Estimation and SlidingMode Control of Markovian Jump Singular Systems’, IEEETransactions on Automatic Control, 55, 1213–1219.

Wu, L., Su, X., and Shi, P. (2012), ‘Sliding Mode Control WithBounded l2 Gain Performance of Markovian Jump SingularTime-delay Systems’, Automatica, 48, 1229–2933.

Yan, X., Lam, J., and Xie, L. (2003), ‘Robust Observer Design forNonlinear Interconnected Systems Using Structural Charac-teristics’, International Journal of Control, 76, 741–746.

Yau, S. (1994), ‘Finite-dimensional Filters With Nonlinear Drift.I: A Class of Filters Including Both Kalman-Bucy and BenesFilters’, Journal of Mathematical Systems, Estimation, andControl, 4, 181–203.

Yoshida, T., and Loparo, K. (1989), ‘Quadratic Regulatory The-ory for Analytic Nonlinear Systems With Additive Controls’,Automatica, 25, 531–544.

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