HOSM Driven Output Tracking in the Nonminimum-Phase Causal Nonlinear Systems

Shtessel1, and Ilia Shkolnikov2 1 ECE department, The University of Alabama in Huntsville, 301 Sparkman Dr., Huntsville, AL, 35899 {baevs,shtessel}@eng.uah.edu 2 Z/I Imaging Corporation,

Nonminimum-Phase Causal Nonlinear Systems

Simon Baev1, Yuri B Shtessel1, and Ilia Shkolnikov2

1 ECE department, The University of Alabama in Huntsville, 301 Sparkman Dr., Huntsville, AL, 35899

{baevs,shtessel}@eng.uah.edu

2

Z/I Imaging Corporation, an Intergraph Company, 230 Business Park Blvd., Madison, AL 35757

ilya.shkolnikov@intergraph.com

Output tracking in causal nonminimum-phase nonlinear systems is a challenging,

real-life control problem In that class of dynamic systems, where the internal

or zero-dynamics are unstable, traditional and powerful control methods such

as feedback linearization [1] and sliding mode control [8, 9] can barely be used Nonminimum-phase output tracking is extensively studied in linear [10] and nonlinear [11, 12] systems A comprehensive review of different design methods for output tracking of nonlinear nonminimum phase systems is given in [12] The output tracking control theory provides a solution for a class of systems where the zero dynamics is described via finite dimensional exosystem In the ideal case with no uncertainties and disturbances, the asymptotic convergence

of the output tracking error is proven in [1] Different aspects of real engineering systems affected by uncertainties or/and disturbances, where even asymptotic convergence can not be guaranteed are presented in [2, 3]

A feedback linearization based output tracking approach is proposed for SISO systems in [4] On the first stage, the high gain feedback linearization is applied

to the system making it linear Then, the stabilization of the internal dynamics

is achieved by using the output variable as a quasi control Finally, the feed-forward inverse of the internal dynamics along with high gain feedback is used

to asymptotically stabilize the system to its equilibrium state The drawback

of such design is in direct stabilizing of the internal dynamics only but not the

whole system instead

Similar approach, based on the feedback linearization is introduced in [5] for nonminimum phase nonlinear MIMO systems Authors propose a two step algorithm which is in linearizing of the I/O dynamics along with splitting of the whole system into two parts on the first step and in design of the separate control law for each part on the second step The whole system is split into generally nonlinear term which represents internal dynamics and a part of I/O dynamics and the rest of the I/O dynamics, which is linear The control law for each part is designed separately: linear state feedback together with feedforward

G Bartolini et al (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp 159–177, 2008 springerlink.com  Springer-Verlag Berlin Heidelberg 2008c

of non-causal inverse for the first part and the linear high gain state feedback for the second one The proposed design has two weak points: the causality level

is weak (non-causal inverse is used in the control design) and it is not robust to the external disturbances

An engineering extension of the method introduced in [5] is studied in [12] Instead of asymptotic output tracking with complete stabilizing of the tracking error, authors consider a zone convergence in the presence of external distur-bances and model uncertainties The method is also based on two steps: sep-arating of the dynamics (linear and nonlinear) with further designing of the control The novelty of the algorithm is in control law design procedure, which does not require feedforwarding of non-causal inverse solution

Robust stabilization for a class of systems with a nonlinear disturbance by means of a dynamical compensator is considered in [13] A trajectory lineariza-tion method for tracking an unstable, nonminimum-phase nonlinear plant is studied in [14] Exact tracking of arbitrary reference signals in causal nonlinear nonminimum-phase systems seems to be difficult to implement even in undis-turbed systems The tracking problem for a class of signals given by a known nonlinear exosystem is reduced to solving a 1storder partial differential-algebraic equation in [1] An approximate solution for such a system for a special class

of systems and trajectories is proposed in [15] Exact tracking of a known tra-jectory given by a noncausal system is achieved via a stable nonlinear inverse

in [16] and is accomplished using the sliding mode control technique in [6] Asymptotic output tracking for a class of nonlinear uncertain systems where the plant is presented in the normal form with internal dynamics expanded in

a power series, and a reference output profile together with unmatched distur-bance are defined by a known linear exosystem, is considered in [17, 18, 19] The condition of the known exogenous system or its characteristic polynomial reduces the causality of the addressed problem

In this work the output tracking is addressed for the nonminimum phase non-linear systems with enhanced causality Similar to the works [17, 18, 19] the output reference profile is supposed to be described by a linear exogenous sys-tem, but in this work the exogenous system can be unknown The characteristic polynomial of the aforementioned exosystem, which is used in the controller design, is estimated on line using the HOSM-based parameter observer The use of the HOSM control technique in this work allows handling the out-put tracking in causal nonminimum phase systems without reducing the relative degree as required in the papers [17, 18, 19, 15] that enhances the controller tracking accuracy

The contribution of this work is in the consistent application of a HOSM approach to the output-tracking problem in a class of nonminimum-phase causal dynamic systems and can be summarized as follows:

(1) Causality improvement

Causality is a very challenging factor in nonminimum phase output tracking Novel design of a generator for the state reference profiles without knowing

the corresponding exogenous system characteristic polynomial significantly increases the overall causality of the method introduced in [17, 18, 19]

(2) Handling arbitrary relative degree

The use of the HOSM control law allows designing the controller in one step, instead of two-step solution as in [17, 18, 19] In other words, the first step

of the controller design — introduction of the pseudo output that reduces the relative degree to one as in [17, 18, 19] — is eliminated in this work

Consider a nonlinear plant model presented in a form of input/output dynamics:

⎛

⎜

⎜

⎝

y (r1 ) 1

y (r2 ) 2

y (r m)

m

⎞

⎟

⎟

and internal dynamics [1]:

˙η = Q η +

m



i=1

where

u{u1, u2, , u m } ∈  mis the control input;

y{y1, y2, , y m } ∈  mis the commanded output (available for measure-ment);

[r1, r2, , r m] ∈  m is the vector relative degree;

r = r1+ r2+· · · + r mis the total relative degree;

η ∈  n −r is the unstable internal dynamics (available for measurement);

n is the total order of the system;

ξ i {y i , ˙ y i , , y (k i)

i } T ∈  k i+1 is the state vector of i thinput-output chan-nel;

ξ {ξ T

1, ξ T2, , ξ T m } T is the combined system state vector;

k i is the order of the highest derivative of i thoutput in the internal dynamics

(k i < r i);

φ( ·){φ1, φ2, , φ m } T ∈  mis a smooth, bounded system function;

f (·) ∈  n −r is a partially defined, smooth enough uncertain term;

Q ∈  (n −r)×(n−r) is the internal dynamics gain matrix (non-Hurwitz);

Gi ∈  (n −r)×(k i+1)is the output gain matrices

Remark 1 The system (1),(2) is nonminimum-phase since the matrix Q is

non-Hurwitz and the internal/zero-dynamics (2) is unstable;

Remark 2 Given in real-time, an output reference profile y c (t) and uncertain term f ( ·) are assumed to be described by unknown linear exosystem of given

order

Remark 3 The emerging HOSM state observation technique for the

nonmini-mum phase system (1),(2) [24] will allow relaxing the assumption of the

mea-suring availability of the internal state η.

The problem is to design a control law u for the causal nonminimum-phase

system (1),(2) that provides asymptotic output tracking of a given in real-time

output reference profile y c {y c1, y c2, , y c m } ∈  m i.e y →y c, as time increases

in the presence of bounded uncertainties and disturbances (both are described

by f ( ·) term).

The problem of nonminimum phase output tracking can be addressed in

dif-ferent ways One of them is in reducing the original problem to a state tracking

problem This can be done by introducing state reference profiles y c and η c The first one is already defined from the original problem formulation The second one is the subject of the stable system center (SSC) approach [17, 18, 19] that

is presented in Sect 4

The robustness of the HOSM control can be employed to implicitly compensate for the uncertain term f ( ·) in (2) But this will require larger control authority

(in physical implementation meaning) that is not acceptable condition for some

cases Therefore, it is useful to estimate such term and explicitly compensate it.

Assume that the partially defined uncertain term f (·) can be presented as a sum

of known and unknown components:

f (·) = f0+ Δf

Assume also that y and η are available for measurement Using (2), the

un-known component Δf can be estimated as follows:

ˆ

Δf = ˆ˙ η − Q η −

m



i=1

Giˆ

i − f0, (3)

where estimates ˆ˙η and ˆ ξ i are subjects of employing the exact higher order sliding

mode differentiator [20, 21] that is considered in more details in Sect 4.1 Those

estimates are available in a finite time

Assume that reference tracking profiles for each command output y i c and

inter-nal dynamics η c are given in real-time They should satisfy the following set of conditions:

C1 y i c is differentiable at least (r i − 1) times;

C2 η c is bounded and satisfies

˙η c = Q η c+

m



i=1

Gi ξ i

where ξ i c {y i c , ˙ y i c , , y (k i)

i c } T

Introduce tracking errors:

e y i (y i c − y i) ∈ ,

e ξ i (ξ i c − ξ i) ∈  k i+1,

e η (η c − η) ∈  n −r .

(5)

Taking p time derivatives of the internal dynamics tracking error e η yields the following equation:

e (p) η = Qp e η+

p



i=1

Qp −i m



j=1

Gj e (i −1)

ξ j +

p



i=1

Qp −i q (i −1) ,

where

qf0(η c , ξ c)− f0(η c − e η , ξ c − e ξ ),

ξ c {ξ T

1c , ξ T2

c , , ξ T m

c } T ,

e ξ {e T

ξ1, e T ξ

2, , e T ξ

m } T

For each input-output pair, define sliding variable as linear combination of

corresponding tracking errors and their derivatives:

⎧

⎨

⎩

σ i e (k i)

y i + Ci e ξ i + T e η

Ci {C i,0 , C i,1 , , C i,k i −1 , 0 } ∈  k i+1

T{T1, T2, , T n −r } ∈  n −r

(6)

where C i,j and T iare coefficients to be designed

Taking (r i −k i ) derivatives of each σ iresults in showing up the corresponding

control u i:

⎧

⎪

⎪

σ (r i −k i)

i = y (r i)

i c − φ i − u i+ Ci e (r i −k i)

ξ i + T e (r i −k i)

η = χ i − u i

χ i y (r i)

i c − φ i+ Ci e (r i −k i)

ξ i + T e (r i −k i)

η

i = 1, m

(7)

The control problem is now decoupled into m independent identical subprob-lems which are in designing m HOSM controls u i for i = 1, m [21].

For a SISO system of the form:

the higher order sliding mode control that provides a finite time stabilization of

σ and its time derivatives up to (r − 1) thorder can be implemented as follows:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

N 1,r =|σ| (r −1)/r

N i,r = |σ| p/r+| ˙σ| p/(r −1) + + |σ (i −1) | p/(r −i+1)(r −i)/p

N r −1,r = |σ| p/r+| ˙σ| p/(r −1) + + |σ (r −2) | p/21/p

ν 0,r = σ

ν 1,r = ˙σ + β1N 1,r sign(σ)

ν i,r = σ (i) + β i N i,r sign(ν i −1,r)

ν r −1,r = σ (r −1) + β

r −1 N r −1,r sign(ν r −2,r)

u = α sign(ν r −1,r)

(9)

where p being the least common multiple of 1, 2, , r and α, β1, , β r −1 are

arbitrary positive parameters (β i < β i+1) to be chosen sufficiently large to

over-come an effect of ψ and provide a finite time convergence of σ.

The existence of the HOSM is proven in [21] The main condition for this is a

boundedness of ψ term in (8) Recalling the original state tracking problem, the

existence condition of HOSM is in boundedness of the collective terms χ iin (7) Such terms are considered to be bounded since they are smooth functions of bounded components:

• Output reference profiles y c i can be designed to be bounded along with its

time-derivatives of any order up to r i;

• The boundedness of internal dynamics profile η c will be proven in the next section;

• Tracking errors e y i , e ξ

i and e ηare bounded because of finite time convergence

of HOSM

In the sliding mode (each σ i= 0), error dynamics is described by the linear

system of order K =

m



i=1

k i + (n − r):

⎧

⎪

⎪

e (k i)

y i =−C i e ξ

i − T e η

˙e η = Q e η+

m



i=1

that can be tuned to have a desirable transient response by selecting K

coeffi-cients of vectors Ci and T.

Remark 4 Since the internal dynamics η are unstable, a general solution η c of (4) may be unbounded This yields unboundedness of| ˙χ| in (7), which makes

the state tracking control u unrealizable In the following section a method of

stable system center [17, 18, 19] is discussed The method allows generation of

a bounded profile ˆη c that asymptotically converges to a bounded solution of

unstable internal dynamics η c i.e ˆη c →η c as time increases Furthermore, the rate of convergence is under control

The method of stable system center (SSC) allows generation of a bounded par-ticular solution ˆη c for the internal dynamics η of system (2) which will converge

to the solution η c of (4) asymptotically as time increases That solution η c is

also known as the ideal internal dynamics (IID) [?, 17, 18, 19].

Introduce the internal dynamics forcing term as the collective excitation func-tion in (4):

θ c(·) 

m



i=1

Gi ξ c

i+ f0+ ˆΔf

 {θ c1, θ c2, , θ c (n −m) } T ∈  n −r

(11)

Assume that θ c(·) is described by some linear exosystem of order k:



˙τ = A τ

with unknown gain matrices A and C and therefore unknown characteristic polynomial P k (λ):

P k (λ) = |A − λ I| = λ k

+ p k −1 λ k −1 + + p

1λ + p0 (13) The characteristic polynomial (13) can be identified in real-time based only

on the knowledge of the order k of system (12) by a HOSM parameter observer

developed in [22]

Polynomial

This method is based on two procedures: exact HOSM differentiation [20, 21] and least-squares estimation [25, 22, 23] that are to be applied to the output θ c

of system (12)

Reducing the Problem to Regressive Form

Consider a linear system in the form (12) The matrices A ∈  k ×k and C ∈

 (n −m)×k are unknown but they have to satisfy the observability condition:

rank(M ) = k, M = {M T

1, , M k T } T ∈  k(n −r)×k ,

An exact HOSM differentiator [20, 21] that is to be applied to θ c j for j =

1, n − r is described as follows:

⎧

⎪

⎪

⎪

⎪

ν 0,j =−λ0|z 0,j − θ c j | n/(n+1) sign(z 0,j − θ c j ) + z 1,j

ν i,j=−λ i |z i,j − ν i −1,j | (k −i)/(k−i+1) sign(z

i,j − ν i −1,j ) + z i+1,j ,

˙z i,j = ν i,j , i = 1, k − 1

˙z k,j=−λ k sign(z k,j − ν k −1,j)

(15)

where the common term z i,j stands for i th derivative of the j thcomponent of the

vector θ c , and the coefficients λ ihave to be selected to guarantee the convergence

of a differentiator [21] Combining z i,j by the i thindex yields the following:

⎧

⎪

⎪

⎪

⎪

Z0{z 0,1 , z 0,2 , , z 0,n −r } T = θ c = C τ

Z1{z 1,1 , z 1,2 , , z 1,n −r } T = ˙θ c = C A τ

Z k −1 {z k −1,1 , z k −1,2 , , z k −1,n−r } T = θ (k −1)

c = C A k −1 τ

Z k {z k,1 , z k,2 , , z k,n −r } T = θ (k) c = C A k τ

(16)

where Z i ∈  n −r corresponds to i th derivative of θ c

Introduce two auxiliary vectors:

Z {Z T

0, Z1T , , Z k T −1 } T and Z¯{Z T

1, Z2T , , Z k T } T (17) which are related through the time derivative ¯Z ≡ ˙Z.

Using (14),(16) and (17) introduce a linear transformation of the state

vector τ :

Introduce an arbitrary, but known matrix D ∈  k ×k (n−r) of rank k

Pre-multiplying both sides of (18) by D, and defining

˜

Z  (D Z) ∈  k , M˜ (D M) ∈  k ×k

where ˜M is assumed to be nonsingular since rank(D) = rank(M ) = k, yields the

following:

˜

Z = ˜ M τ ∴ τ = ˜ M −1 Z˜ Taking the derivative of both sides, the dynamics of system, similar to (12), can be derived as follows:

˙˜

Z = ˜ M ˙τ = ˜ M A τ = ˜ M A ˜ M −1

  

˜

A

˜

Z = ˜ A ˜ Z (19)

Recalling that ˜Z = D Z and ˙ Z = ¯ Z gives the way to express Z:˙˜

˙˜

Z = D ¯ Z

Putting it all together allows treating (19) as a set of k linear expressions in the regressive form:

H = K Q − equation in the regressive form;

H ˙˜Z = D ¯ Z − known left-hand side vector;

Q ˜Z = D Z − known right-hand side vector;

K ˜A − unknown matrix to be identified;

(20)

or in the scalar notation:

H i=

k



j=1

Unknown coefficients K i,j for i, j = 1, k in scalar equations (21) are to be identified via the least-square estimation method, which is presented in the next

subsection As soon as the matrix ˜A ≡K is estimated, its characteristic

poly-nomial can easily be computed Since ˜A and A are similar due to eq.(19) the

characteristic polynomials of the matrices are the same

Least-Square Estimation (LSE) Method

Single variable identification

Consider a scalar linear equation:

where k is the constant coefficient to be identified; q(t) and h(t) are known signals The unknown parameter k can not be determined uniquely, since values

of functions q(t) and h(t) do not necessary satisfy condition (22) for all time

moments

Multiplying (22) by q(t) and integrating both parts from some initial time moment t0to current time gives the following:

k

t



t0

q(τ )2dτ =

t



t0

q(τ ) h(τ ) dτ

which yields a way to determine the unknown scalar coefficient k in real time starting from moment t0:

k =

t



t0

q(τ ) h(τ ) dτ

t



t0

q(τ )2dτ

(23)

Multivariable identification

Consider a linear equation that fits a regressive form of the order k:

h(t) = k1q1(t) + k2q2(t) + + k k q k (t) where q i (t) and h(t) are known time functions (with values which are measured

or computed) and the {k1, k2, , k k } is a vector of unknown constants to be

identified

In fact, there are k unknowns and only one equation, thus, this problem can

not be solved uniquely Since q i (t) and h(t) are time functions, therefore, the

following equations are guaranteed:

⎧

⎪

⎨

⎪

⎩

h(t) = q1(t) k1+ + q k (t) k k

h(t + Δ) = q1(t + Δ) k1+ + q k (t + Δ) k k

h(t + (k − 1) Δ) = q1(t + (k − 1) Δ) k1+ + q k (t + (k − 1) Δ) k k

where Δ is some constant time interval.

All of these equations can be grouped into a k-order linear algebraic system:

⎡

⎢

⎢

q 1,1 q 1,2 q 1,k

q 2,1 q 2,2 q 2,k

. . .

q k,1 q k,2 q k,k

⎤

⎥

⎥

⎡

⎢

⎢

k1

k2

k k

⎤

⎥

⎥=

⎡

⎢

⎢

h1

h2

h k

⎤

⎥

where the following notations are used:

q i,j ≡ q j (t + (i − 1) Δ), h i ≡ h(t + (i − 1) Δ)

Each unknown k ican be found, for instance, by means of Kramer’s rule:

k j =











q 1,1 q 1,2 (q 1,j → h1) q 1,k

q 2,1 q 2,2 (q 2,j → h2) q 2,k

. . . . .

q k,1 q k,2 (q k,j → h k ) q k,k





















q 1,1 q 1,2 q 1,j q 1,k

q 2,1 q 2,2 q 2,j q 2,k

. . . . .

q k,1 q k,2 q k,j q k,k











(25)

Since q i,j and h iare time functions, they form time functions in the numerator and denominator of (25) Therefore, this equation can can be rewritten as follows:

k j = Q j (t)

Q(t) , j = 1, k