Flatness, Backstepping and Sliding Mode Controllers for Nonlinear Systems

A backstepping method for designing an SMC for a class of nonlinear systemwithout uncertainties, has been presented by Rios-Bol´ıvar and Zinober [16, 17].The adaptive sliding backsteppin

Controllers for Nonlinear Systems

Ali J Koshkouei1, Keith Burnham1, and Alan Zinober2

1 Control Theory and Applications Centre, Coventry University,

H ∞, proportional-integral-derivative (PID) and self-tuning Note that PID

con-trol design techniques may also be used for designing the sliding surface Adrawback of the SMC methods may be unwanted chattering resulting from dis-continuous control There are many methods which can be employed to reducechattering, for example, using a continuous approximation of the discontinuouscontrol, and a combination of continuous and discontinuous sliding mode con-trollers Chattering may also be reduced using the higher-order SMC [4] anddynamic sliding mode control [4, 5]

When plants include uncertainty with a lack of information about the bounds

of unknown parameters, adaptive control is more convenient; whilst, if sufficientinformation about the uncertainty, such as upper bound is available, a robustcontrol is normally designed The stabilisation problem has been studied for dif-ferent classes of systems with uncertainties in recent years [6]-[10] Most controldesign approaches are based upon Lyapunov and linearisation methods In theLyapunov approach, it is very difficult to find a Lyapunov function for designing

a control and stabilising the system The linearisation approach yields local bility The backstepping approach presents a systematic method for designing acontrol to track a reference signal by selecting an appropriate Lyapunov functionand changing the coordinates [11, 12] The robust output tracking of nonlinearsystems has been studied by many authors [13]-[15] Backstepping techniqueguarantees global asymptotic stability Adaptive backstepping algorithms have

sta-G Bartolini et al (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp 269–290, 2008 springerlink.com  Springer-Verlag Berlin Heidelberg 2008c

been applied to systems which can be transformed into a triangular form, inparticular, the parametric pure feedback (PPF) form and the parametric strictfeedback (PSF) form [12] This method has been studied widely in recent years[11, 12], [15]-[19].

If a plant has matched uncertainty, a state feedback control may stabilise thesystem [7] Many techniques have been proposed for the case of plants contain-ing unmatched uncertainty [20] The plant may contain unmodelled terms andunmeasurable external disturbances bounded by known functions or their norm

is bound to a constant

SMC is a robust control method and backstepping can be considered to be

a method of adaptive control The combination of these methods, the so-calledadaptive backstepping SMC, yields benefits from both approaches This methodcan be used even if the system does not comprise of an unknown parameter Thebackstepping sliding mode approach has been extended to some classes of non-linear systems which need not be in the PPF or PSF forms [15]-[19] A symbolicalgebra toolbox allows straightforward design of dynamical backstepping control[16] A backstepping method for designing an SMC for a class of nonlinear systemwithout uncertainties, has been presented by Rios-Bol´ıvar and Zinober [16, 17].The adaptive sliding backstepping control of semi-strict feedback systems (SSF)[21] has been studied by Koshkouei and Zinober [22]

In this chapter, a systematic design procedure is proposed to combine adaptivecontrol and SMC techniques for a class of nonlinear systems In fact, the back-stepping approach for SSF systems with unmatched uncertainty is developed Acontroller based on SMC techniques is designed so that the state trajectories ap-proach a specified hyperplane These systematic methods do not need any extracondition on the parameters and also any sufficient conditions for the existence

of the sliding mode to guarantee the stability of the system

On the other hand, flatness is an important property in control theory whichassures that the system can be stabilised by imposing an artificial output [23]-[25] A linear system is flat if and only if it is controllable A SISO system with

an output is not flat if the relative degree of the system with respect to theoutput (if it is defined and finite) is not the same as the order of the system

In general, there is no comprehensive systematic method for classifying flat andnon-flat systems, and also for finding a suitable flat output for nonlinear systems.However, the controllability matrix yields a flat output for a linear system [23]and flat time-varying linear systems have been studied by Sira-Ram´ırez andSilva-Navarro [26] In addition, the control of non-flat systems is an importantissue which has been studied since the last decade [24, 27, 28]

Flat outputs may not be the actual outputs of the system Flatness for thetracking problem of linear systems in differential operator representation hasbeen considered by Deutscher [29] For MIMO nonlinear systems, there are dif-ferences between exact feedback linearisablility and differential flatness (for ex-ample see [24, 28]) However, most published papers have dealt with flatness ornon-flatness of SISO systems

Exact feedforward linearisation based upon differential flatness has been ied by Hagenmeyer and Delaleau [30] in which a flat system is linearised viafeedforward control using the differential flatness trajectory satisfying a certaincondition on the initial conditions In fact there is a relationship between the flat-ness and linearisability of nonlinear systems by feedback In particular, for singleinput systems, flatness is equivalent to linearisability by static state feedbackand static feedback linearisability is equivalent to dynamic feedback linearisabil-ity [31] In other words, linearisation via static (dynamic) state feedback andcoordinate transformation is equivalent to the linearisation by the static (dy-namics) feedback of some outputs and a finite number of their derivatives Thepractical and asymptotic tracking problems for nonlinear systems when only theoutput of the plant and the reference signal are available has been considered in[32] In addition the concept of global flatness has been presented A system isnot globally flat if either the relative degree of the associated augmented system

stud-is not well-defined everywhere or the change of coordinates using a particulartransformation is not a global diffeomorphism [32]

SMC and second-order SMC for nonlinear flat systems are also considered

in this chapter The method benefits from the advantages of both approaches.The important and main property of SMC is its robustness in the presence ofmatched uncertainties whilst the flatness property guarantees that the controlcan be obtained as a function of the flat output and its derivatives In thiscase, the sliding surface is also introduced in terms of the flat output and itsderivatives

Differential flatness property and the second-order SMC for a hovercraft vesselmodel has been studied in [33] The technique has been proposed for the spec-ification of a robust dynamic feedback multivariable controller accomplishingprescribed trajectory tracking tasks for the earth coordinate position variables.Moreover, in this chapter a gravity-flow tank/pipeline system is stabilised via

an SMC obtained from flatness and sliding mode control theory This combinedmethod inherits the robustness property from SMC If sufficient informationabout the flat output is available then the control is accessible and applicablewithout requiring further knowledge of the system variables

This chapter is organised as follows: The classical backstepping method tocontrol systems in the parametric semi-strict feedback form is extended in Sec-tion 2 to achieve the output tracking of a dynamical reference signal The SMCdesign based upon the backstepping approach is presented in Section 3 An ex-ample which illustrates the results of the backstepping method, is presented inSection 4 In Section 5 the definition and properties of flatness for nonlinearsystems are considered In Section 6 a control design method for a class of non-linear systems with unknown parameters using SMC and the flatness techniques

is proposed A suitable estimate for unknown parameters is also obtained InSection 7 the SMC flatness results are applied to a gravity-flow tank/pipelinemodel for controlling the volumetric flow rate of the liquid leaving the tankand the height of the liquid in the tank presented Conclusions are given inSection 8

2 Adaptive Backstepping Control

In this section the backstepping procedure for a class of nonlinear systems withunmatched disturbances is presented Consider the uncertain system

˙χ = F (χ) + G(χ)θ + Q(χ)u + D(χ, w, t) (1)

where χ ∈ R n is the state and u the scalar control The functions F (χ) ∈ R n,

G(χ) ∈ R n ×p and Q(χ) ∈ R n are known D(χ, w, t) ∈ R n and w are unknown function and an uncertain time-varying parameter, respectively Also θ ∈ R p

is the vector of constant unknown parameters Assume that the system (1) istransformable into the semi-strict feedback form (SSF) [21, 22, 34]

where x = [x1 x2 x n]T is the state, y the output, f n (x), g n (x) ∈ R and

ϕ i (x1, , x i) ∈ R p , i = 1, , n, are known functions which are assumed to

be sufficiently smooth η i (x, w, t), i = 1, , n, are unknown nonlinear scalar

functions including all the disturbances

Assumption 1 The functions η i (x, w, t), i = 1, , n are bounded by known positive functions h i (x1, x i)∈ R, i.e.

|η i (x, w, t) | ≤ h i (x1, x i ), i = 1, , n (3)

The output y should track a specified bounded reference signal y r (t) with bounded derivatives up to the n-th order.

The system (1) is transformed into system (2) if there exists an appropriate

diffeomorphism x = x(χ) The conditions of the existence of a diffeomorphism

x = x(χ) can be found in [35] and the input-output linearisation results in [36].

First, a classical backstepping method will be extended to this class of systems

to achieve the output tracking of a dynamical reference signal The SMC designbased upon backstepping techniques is then presented in Section 3

2.1 Backstepping Algorithm

The design method based upon the adaptive backstepping approach has beenpresented in [22, 34] and is recalled afterwards This method ensures that theoutput tracks a desired reference signal

Step 1. Define the error variable z1= x1− y rthen

˙z1= x2+ ϕ T (x1)θ + η (x, w, t) − ˙y r (4)

z1=−c1z1+ z2+ ω T1˜θ + η1(x, w, t) − n h21z1e at (10)and ˙V1 is converted to

3 Sliding Mode Backstepping Controllers

When there are uncertainties in the system, adaptive control or SMC techniquesmay be used to design an appropriate controller SMCs are insensitive withrespect to matched uncertainties However, SMCs may reduce the effect of un-matched disturbances significantly A robust control for a plant with uncertainty

may be obtained using a combined method of SMC and adaptive control niques A combination of these methods has been studied in recent years [15]-[19].The adaptive backstepping SMC of SSF systems has been studied by Koshk-ouei and Zinober [22, 34] The controller is based upon SMC and backsteppingtechniques so that the state trajectories approach a specified hyperplane withoutrequiring any sufficient condition for the existence of the sliding mode.

tech-To provide robustness, the adaptive backstepping algorithm can be fied to yield an adaptive sliding output tracking controller The modification iscarried out at the final step of the algorithm by incorporating an appropriatesliding surface defined in terms of the error coordinates The sliding surface isdefined as

where k i > 0, i = 1, , n − 1, are real numbers In addition, the Lyapunov

function (20) is modified as follows

k i ω i

= Γ

n−1 i=1

since from (25), z n = σ − k1z1− k2z2− − k n −1 z n −1 Setting ˙ˆθ = τ n, ˜θ is

eliminated from the right-hand side of (28) Consider the adaptive sliding modeoutput tracking control

guar-between two sliding mode gains W and K which may reduce the chattering

ob-tained from discontinuous term and the desired performances may be achieved

If K is very large with respect o W , unwanted chattering is produced If K is sufficiently large, one can select W so that stability with a significant chattering reduction is established W also affects the reaching time of the sliding mode.

In fact by increasing the value W , the reaching time is decreased.

Remark 1 Alternatively, at the n-th step, one can select the following control

t Control action

Fig 1 Regulator responses with nonlinear control (36) for PSF system

Then the control law (23) becomes

cos(3x1x2) Alternatively, one can design an appropriate SMC for the system

Assume that the sliding surface is σ = k1z1+ z2= 0 with k1> 0 The adaptive

tran-sliding mode and k1 = 1, K = 10, W = 0 The simulation results with K = 10,

W = 5, are shown in Fig 3 If W > 0 the chattering of the sliding motion is

t Control action

Fig 2 Tracking responses with sliding control (37) for PSF system with K = 10 and

W = 0

reduced and also the reaching time is shorter than when w = 0 So trade off for

a suitable selection of the gain pair K and W is an important issue which may

affect the chattering

5 Flatness

As stated, there is a link between the differential flatness and the feedback earisation problem If the derivative of the state can be expressed in terms ofthe system state and the derivatives of input variables then the state is calledthe generalised state and the preceding equations are referred to as a generalisedstate representation of the system [37] If the generalised state representationsare used for designing a feedback control, the time derivatives of the input vari-ables may appear in the feedback laws This feedback is known as a quasi-staticstate feedback (see [38] and references therein) A flat nonlinear system is lin-earisable via a generalised quasi-static state feedback For SISO systems, thelinearisability and flatness properties are equivalent Therefore the control ob-tained stabilises the systems without including any extra dynamics If the system

t Control action

Fig 3 Tracking responses with sliding control (37) for PSF system with K = 10 and

W = 5

includes uncertainties, particularly matched uncertainties, sliding mode control

is an appropriate approach to achieve the system tracking stability Backsteppingmethod is applicable to minimum-phase nonlinear systems [15] with unknownparameters and disturbances In particular, systems in the form of SFF canbenefit from this technique

Flatness is a geometric system property which does not change the coordinatesand indicates that the system is transformable to an associated linear system.Therefore, a flat system has a well-structured system which enables one to design

a controller and solve the stabilisation problem One can also use the dynamicfeedback linearisation method for control of flat systems However, backsteppingmethod is applicable for a wide class of nonlinear systems Note that there is nosystematic method for constructing a flat output To study the performance offlatness, the definition of flatness is first considered

Definition 1 [30] Consider the nonlinear system

˙

where x ∈ R n is the state, t ∈ R, f(x, u) ∈ R n is a smooth vector field and

u ∈ R m is the control The system (38) is (differentially) flat if there exists a