HOSM Driven Output Tracking in the Nonminimum-Phase Causal Nonlinear Systems Simon Baev1 , Yuri B Shtessel1 , and Ilia Shkolnikov2 ECE department, The University of Alabama in Huntsville, 301 Sparkman Dr., Huntsville, AL, 35899 {baevs,shtessel}@eng.uah.edu Z/I Imaging Corporation, an Intergraph Company, 230 Business Park Blvd., Madison, AL 35757 ilya.shkolnikov@intergraph.com Introduction 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 feedforward 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 c Springer-Verlag Berlin Heidelberg 2008 springerlink.com 160 S Baev, Y.B Shtessel, and I Shkolnikov 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 disturbances and model uncertainties The method is also based on two steps: separating 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 linearization 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 undisturbed systems The tracking problem for a class of signals given by a known nonlinear exosystem is reduced to solving a 1st order 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 trajectory 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 disturbance 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 nonlinear systems with enhanced causality Similar to the works [17, 18, 19] the output reference profile is supposed to be described by a linear exogenous system, 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 output 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 HOSM Driven Output Tracking 161 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 Problem Formulation 2.1 Original Output Tracking Problem Consider a nonlinear plant model presented in a form of input/output dynamics: ⎛ (r ) ⎞ y1 ⎜ (r2 ) ⎟ ⎜ y2 ⎟ ⎜ ⎟ = φ(ξ, η, t) + u(ξ, η), (1) ⎜ ⎟ ⎝ ⎠ (r ) ym m and internal dynamics [1]: m η = Qη + ˙ Gi ξ i + f (ξ, η, t), (2) i=1 where u {u1 , u2 , , um } ∈ m is the control input; y {y1 , y2 , , ym } ∈ m is the commanded output (available for measurement); [r1 , r2 , , rm ] ∈ m is the vector relative degree; r = r1 + r2 + · · · + rm is the total relative degree; η ∈ n−r is the unstable internal dynamics (available for measurement); n is the total order of the system; (k ) ξ i {yi , yi , , yi i }T ∈ ki +1 is the state vector of ith input-output chan˙ nel; ξ {ξ T , ξ T , , ξ T }T is the combined system state vector; m ki is the order of the highest derivative of ith output in the internal dynamics (ki < ri ); φ(·) {φ1 , φ2 , , φm }T ∈ m is 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)×(ki +1) is the output gain matrices Remark The system (1),(2) is nonminimum-phase since the matrix Q is nonHurwitz and the internal/zero-dynamics (2) is unstable; 162 S Baev, Y.B Shtessel, and I Shkolnikov Remark Given in real-time, an output reference profile yc (t) and uncertain term f (·) are assumed to be described by unknown linear exosystem of given order Remark The emerging HOSM state observation technique for the nonminimum phase system (1),(2) [24] will allow relaxing the assumption of the measuring 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 yc {yc1 , yc2 , , ycm } ∈ m i.e y→yc , 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 different ways One of them is in reducing the original problem to a state tracking problem This can be done by introducing state reference profiles yc 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 State Tracking HOSM Control Design 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 3.1 The Disturbance Estimation 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 unknown component Δf can be estimated as follows: m Gi ˆi − f0 , ξ ˆ ˆ Δf = η − Q η − ˙ (3) i=1 ˆ where estimates η and ˆi are subjects of employing the exact higher order sliding ˙ ξ mode differentiator [20, 21] that is considered in more details in Sect 4.1 Those estimates are available in a finite time 3.2 State Tracking Assume that reference tracking profiles for each command output yic and internal dynamics η c are given in real-time They should satisfy the following set of conditions: HOSM Driven Output Tracking 163 C1 yic is differentiable at least (ri − 1) times; C2 η c is bounded and satisfies m ˆ Gi ξ ic + f0 + Δf , η c = Q ηc + ˙ (4) i=1 (k ) ˙ where ξ ic {yic , yic , , yic i }T Introduce tracking errors: eyi (yic − yi ) ∈ eξi (ξ ic − ξ i ) ∈ eη (η c − η) ∈ , ki +1 n−r (5) , Taking p time derivatives of the internal dynamics tracking error eη yields the following equation: p e(p) = Qp eη + η i=1 where p m (i−1) Qp−i Gj eξj Qp−i q (i−1) , + j=1 i=1 q f0 (η c , ξ c ) − f0 (η c − eη , ξ c − eξ ), ξ c {ξ Tc , ξ Tc , , ξ T c }T , m eξ {eT1 , eT2 , , eTm }T ξ ξ ξ For each input-output pair, define sliding variable as linear combination of corresponding tracking errors and their derivatives: ⎧ (k ) ⎨ σ i e y i i + Ci e ξ i + T e η (6) C {Ci,0 , Ci,1 , , Ci,ki −1 , 0} ∈ ki +1 ⎩ i T {T1 , T2 , , Tn−r } ∈ n−r where Ci,j and Ti are coefficients to be designed Taking (ri − ki ) derivatives of each σ i results in showing up the corresponding control ui : ⎧ (r −k ) (r ⎪ σ i i i = yic i ) − φi − ui + Ci e(ri −ki ) + T e(ri −ki ) = χi − ui η ⎨ ξi (r ) (r −k ) (r −k ) (7) χi yic i − φi + Ci eξi i i + T eη i i ⎪ ⎩ i = 1, m The control problem is now decoupled into m independent identical subproblems which are in designing m HOSM controls ui for i = 1, m [21] 3.3 An Arbitrary-Order Sliding Controller For a SISO system of the form: σ (r) = ψ − u (8) 164 S Baev, Y.B Shtessel, and I Shkolnikov the higher order sliding mode control that provides a finite time stabilization of σ and its time derivatives up to (r − 1)th order can be implemented as follows: ⎧ (r−1)/r ⎪ N1,r = |σ| ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ N = |σ|p/r + |σ|p/(r−1) + + |σ (i−1) |p/(r−i+1) (r−i)/p ⎪ i,r ˙ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Nr−1,r = |σ|p/r + |σ|p/(r−1) + + |σ (r−2) |p/2 1/p ˙ ⎪ ⎨ ν 0,r = σ (9) ⎪ ν 1,r = σ + β N1,r sign(σ) ˙ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ν i,r = σ (i) + β i Ni,r sign(ν i−1,r ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ν ⎪ r−1,r = σ (r−1) + β r−1 Nr−1,r sign(ν r−2,r ) ⎪ ⎩ u = α sign(ν r−1,r ) where p being the least common multiple of 1, 2, , r and α, β , , β r−1 are arbitrary positive parameters (β i < β i+1 ) to be chosen sufficiently large to overcome 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 χi in (7) Such terms are considered to be bounded since they are smooth functions of bounded components: • Output reference profiles yci can be designed to be bounded along with its time-derivatives of any order up to ri ; • The boundedness of internal dynamics profile η c will be proven in the next section; • Tracking errors eyi , eξi and eη are bounded because of finite time convergence of HOSM In the sliding mode (each σ i = 0), error dynamics is described by the linear m ki + (n − r): system of order K = i=1 ⎧ (k ) ⎪ eyi i = −Ci eξi − T eη ⎨ m ⎪ eη = Q eη + ⎩˙ Gi eξi + q (10) i=1 that can be tuned to have a desirable transient response by selecting K coefficients of vectors Ci and T Remark 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 ˙ HOSM Driven Output Tracking 165 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 η c that asymptotically converges to a bounded solution of ˆ unstable internal dynamics η c i.e η c →η c as time increases Furthermore, the ˆ rate of convergence is under control Stable System Center Design The method of stable system center (SSC) allows generation of a bounded particular solution η c for the internal dynamics η of system (2) which will converge ˆ 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 function in (4): m ˆ Gi ξ ci + f0 + Δf θc (·) (11) i=1 {θc1 , θc2 , , θc(n−m) }T ∈ n−r Assume that θc (·) is described by some linear exosystem of order k: τ = Aτ ˙ θc = C τ (12) with unknown gain matrices A and C and therefore unknown characteristic polynomial Pk (λ): Pk (λ) = |A − λ I| = λk + pk−1 λk−1 + + p1 λ + 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] 4.1 Identification of the Linear Exosystem’s Characteristic 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 θc 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: T T rank(M ) = k, M = {M1 , , Mk }T ∈ i−1 (n−r)×k ∈ , i = 1, k Mi = C A k(n−r)×k , (14) 166 S Baev, Y.B Shtessel, and I Shkolnikov An exact HOSM differentiator [20, 21] that is to be applied to θcj for j = 1, n − r is described as follows: ⎧ ⎪ ν 0,j = −λ0 |z0,j − θ cj |n/(n+1) sign(z0,j − θ cj ) + z1,j ⎪ ⎪ ⎪ ⎪ ⎨ (k−i)/(k−i+1) (15) sign(zi,j − ν i−1,j ) + zi+1,j , ⎪ ν = −λi |zi,j − ν i−1,j | ⎪ i,j ⎪ zi,j = ν i,j , i = 1, k − ⎪˙ ⎪ ⎩ zk,j = −λk sign(zk,j − ν k−1,j ) ˙ where the common term zi,j stands for ith derivative of the j th component of the vector θc , and the coefficients λi have to be selected to guarantee the convergence of a differentiator [21] Combining zi,j by the ith index yields the following: ⎧ T ⎪ Z0 {z0,1 , z0,2 , , z0,n−r } = θc = C τ ⎪ ⎪ Z {z , z , , z T ˙ ⎪ ⎪ 1,1 1,2 1,n−r } = θ c = C A τ ⎨ ⎪ ⎪ ⎪Z ⎪ k−1 {zk−1,1 , zk−1,2 , , zk−1,n−r }T = θ(k−1) = C Ak−1 τ ⎪ c ⎩ Zk {zk,1 , zk,2 , , zk,n−r }T = θ(k) = C Ak τ c (16) where Zi ∈ n−r corresponds to ith derivative of θc Introduce two auxiliary vectors: T T T Z {Z0 , Z1 , , Zk−1 }T T T T ¯ and Z {Z1 , Z2 , , Zk }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 : Z = M τ, (18) Introduce an arbitrary, but known matrix D ∈ multiplying both sides of (18) by D, and defining ˜ Z (D Z) ∈ k , ˜ M k×k (n−r) (D M ) ∈ of rank k Pre- k×k ˜ where M is assumed to be nonsingular since rank(D) = rank(M ) = k, yields the following: ˜ ˜ ˜ ˜ Z=Mτ ∴ 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 Z = A Z ˜ A ˙ ˜ ˙ ¯ ˜ Recalling that Z = D Z and Z = Z gives the way to express Z: ˙ ˜ ¯ Z = DZ (19) HOSM Driven Output Tracking 167 Putting it all together allows treating (19) as a set of k linear expressions in the regressive form: − equation in the regressive form; H =KQ ˙ ˜ ¯ H Z = DZ ˜ Q Z = DZ ˜ K A − known left-hand side vector; − known right-hand side vector; − unknown matrix to be identified; or in the scalar notation: (20) k Hi = Ki,j Qj (21) j=1 Unknown coefficients Ki,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: h(t) = k q(t) (22) 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) not necessary satisfy condition (22) for all time moments Multiplying (22) by q(t) and integrating both parts from some initial time moment t0 to current time gives the following: t t k q(τ ) dτ = t0 q(τ ) h(τ ) dτ t0 which yields a way to determine the unknown scalar coefficient k in real time starting from moment t0 : t q(τ ) h(τ ) dτ k= t0 (23) t q(τ ) dτ t0 168 S Baev, Y.B Shtessel, and I Shkolnikov Multivariable identification Consider a linear equation that fits a regressive form of the order k: h(t) = k1 q1 (t) + k2 q2 (t) + + kk qk (t) where qi (t) and h(t) are known time functions (with values which are measured or computed) and the {k1 , k2 , , kk } 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 qi (t) and h(t) are time functions, therefore, the following equations are guaranteed: ⎧ ⎪ h(t) = q1 (t) k1 + + qk (t) kk ⎪ ⎪ ⎨ h(t + Δ) = q1 (t + Δ) k1 + + qk (t + Δ) kk ⎪ ⎪ ⎪ ⎩ h(t + (k − 1) Δ) = q1 (t + (k − 1) Δ) k1 + + qk (t + (k − 1) Δ) kk where Δ is some constant time interval All of these equations can be grouped into a k-order linear algebraic system: ⎤⎡ ⎤ ⎡ ⎤ ⎡ k1 h1 q1,1 q1,2 q1,k ⎢q2,1 q2,2 q2,k ⎥ ⎢ k2 ⎥ ⎢h2 ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎢ (24) ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎣ ⎦⎣.⎦ ⎣.⎦ qk,1 qk,2 qk,k kk hk where the following notations are used: qi,j ≡ qj (t + (i − 1) Δ), hi ≡ h(t + (i − 1) Δ) Each unknown ki can be found, for instance, by means of Kramer’s rule: q1,1 q1,2 (q1,j → h1 ) q1,k q2,1 q2,2 (q2,j → h2 ) q2,k qk,1 qk,2 (qk,j → hk ) qk,k kj = q1,1 q1,2 q1,j q1,k q2,1 q2,2 q2,j q2,k (25) qk,1 qk,2 qk,j qk,k Since qi,j and hi are time functions, they form time functions in the numerator and denominator of (25) Therefore, this equation can can be rewritten as follows: kj = Qj (t) , Q(t) j = 1, k HOSM Driven Output Tracking 169 By assuming Q(t)=0 the following equality Q(t) kj = Qj (t) holds and can be addressed by the scalar parameter identification problem considered above: T2 Q(τ ) Qj (τ ) dτ kj = T1 , T2 j = 1, k (26) Q(τ )2 dτ T1 The starting time T1 should be selected to satisfy T1 > (t0 + k Δ) and T2 could be selected as the current time moment t (i.e simulation time) There exists a so-called sliding window that initially takes data samples corresponding to k time moments from (T1 − k Δ) to T1 Then, after the calculation of all ki for this initial data set is done, another portion of data samples (assuming the simulation step size of Δ) corresponding to the interval from (T1 − (k − 1) Δ) to (T1 + Δ) can be placed into the window to get a new set of values ki and so on In order to avoid division by 0, the initial condition for the second integral (denominator) should not be equal to zero Using the LSE Method for Identifying Characteristic Polynomial Coefficients Recalling the original problem of characteristic polynomial identification, the multivariable case, studied in the previous subsection, should be applied k times ˜ (once for each scalar equation (21) with i = 1, k) to estimate matrix A = K = {Ki,j } for i, j = 1, k Then, using the calculated estimate, it is obvious to identify the characteristic polynomial: ˜ Pk (λ) = λ I − A = λk + pk−1 λk−1 + + p1 λ + p0 of the system (19) and, therefore, for the system (12), because of their similarity 4.2 Reference Profile η c Generation Algorithm ˆ Since the characteristic polynomial (13) of system (12) has been identified, the stable system center design procedure can be applied to find a bounded reference profile of the internal dynamics η c The following theorem presents such ˆ procedure Theorem Given the nonminimum phase system (1) and (2) with the measurable state-vector (η, y) and the following set of conditions [17, 18, 19]: Matrix Q in (2) is nonsingular; The output reference profile yc and uncertain term f (·) can be piece-wise modeled by a linear exosystem with unknown characteristic polynomial that is identified in (Sect 4.1) via a HOSM parameter observer 170 S Baev, Y.B Shtessel, and I Shkolnikov Then The real-time output tracking problem of the bounded reference profile yc ∈ m can be replaced by tracking the state-reference profile (ˆc , yc )∈ n such that η (η, y)→(ˆc , yc ) asymptotically as time increases; η The bounded internal state reference profile η c ∈ n−m is generated by a maˆ trix differential equation: ˙ ˆ η (k) + ck−1 η (k−1) + + c1 η c + c0 η c = ˆc ˆc ˆ (k−1) ˙ c + P0 θc + + P1 θ = − Pk−1 θ (27) c where the numbers ck−1 , , c1 , c0 are chosen to provide a desirable eigenvalue placement of convergence η c →η c and matrices ˆ Pk−1 , , P1 , P0 ∈ (n−r)×(n−r) are given by: Pk−1 = I + ck−1 Q−1 + + c0 Q−k × −1 −I × I + pk−1 Q−1 + + p0 Q−k Pk−2 = ck−2 Q−1 + + c0 Q−(k−1) − (Pk−1 + I) × × pk−2 Q−1 + + p0 Q−(k−1) (28) P1 = c1 Q−1 + c0 Q−2 − (Pk−1 + I) (p1 Q−1 + p0 Q−2 ) P0 = c0 Q−1 − (Pk−1 + I) p0 Q−1 where pk−1 , , p1 , p0 are coefficients of characteristic polynomial (13) Proof The proof is given in [17, 18, 19] Theorem gives the way of determining bounded reference profile η c that will ˆ asymptotically track the IID η c in real time with any given convergence rate defined by desirable eigenvalue placement Example The developed HOSM-based nonminimum-phase output tracking algorithm for a class of reference signals with unknown dynamics is applicable to many reallife problems, including DC-to-DC power converters [26, 27, 28] and aircraft control [29, 30] A tutorial illustrative example is considered to illustrate the performance of developed HOSM-based control algorithm The plant’s model is given in form (1),(2): ⎧ (2) ⎪ y1 = y1 − sin(y2 ) + cos(3 t) + u1 ˙ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ y (2) = −2 y y − y + u 2 2 (29) ⎪ η = η + 0.1 y − 0.35 y ⎪ ˙1 ⎪ 2 ⎪ ⎪ ⎪ ⎩ η = −0.15 η1 + 0.8 η − 0.2 y1 − 0.41 y2 ˙ HOSM Driven Output Tracking 171 Fig The tracking performance of the first output Fig The tracking performance of the second output where yi , ui , η i ∈ The total system order n is Relative degree of both outputs with respect to control input is The unstable internal dynamics is presented in linearized form and has non-Hurwitz gain matrix Q with eigenvalues 0.3 and 0.5 Assume that the reference profiles for both outputs are given in real-time and corresponds to the output of the unknown linear exogenous system of the forth order For the purposes of simulation we define output references as oscillating functions: 172 S Baev, Y.B Shtessel, and I Shkolnikov Fig The tracking performance of the first internal state y1c = 1.237 sin(3.05 t), t < 60 , 2.576 sin(2.17 t), t≥60 y2c = 2.576 sin(2.17 t), t