Preprints of the 8th IFAC Symposium on Advanced Control of Chemical Processes The International Federation of Automatic Control Furama Riverfront, Singapore, July -3, Set-based adaptive estimation for a class of nonlinear systems with time-varying parameters Samandeep Dhaliwal and Martin Guay Department of Chemical Engineering, Queen s University, Kingston, ON, Canada E-mail address: martinguay@cheequeensuca Abstract: In this paper, an adaptive estimation technique is proposed for the estimation of time-varying parameters for a class of continuous-time nonlinear system A set-based adaptive estimation is used to estimate the time-varying parameters along with an uncertainty set The proposed method is such that the uncertainty set update is guaranteed to contain the true value of the parameters Unlike existing techniques that rely on the use of polynomial approximations of the time-varying behaviour of the parameters, the proposed technique does require a functional representation of the time-varying behaviour of the parameter estimates A simulation example is used to illustrate the developed procedure and ascertain the theoretical results Keywords: Nonlinear estimation, adaptive estimation, time-varying parameters INTRODUCTION The problem of parameter estimation has been of considerable interest during the last two decades The vast majority of the literature on adaptive estimation and adaptive control relies on the assumption that the parameters of the system are slowly time-varying or constant In practice however, the time-varying behaviour of the process parameters may be of significant importance -varying parameters can arise from uncertain complex mechanisms not captured by the process model They can also arise from model-plant mismatch or unmeasured inputs that affect the process dynamics Evidently, the ability to estimate such time-varying behaviour can have a significant impact on control system performance Several researchers have considered the problem of estimation of time-varying parameters In Zhu and Pagilla [6], an estimation routine is proposed for a class of nonlinear systems with time-varying parameters A Taylor series expansion of the time-varying parameters is considered The time varying behaviour is captured by applying a local polynomial expansion and estimating the locally constant Taylor series coefficients using a standard least-squares approach The authors demonstrate the convergence of the parameter estimates to a neighbourhood of the true parameters within a given resetting period An related approach is proposed in Chen et al [] where a direct least-squares based method is used to estimate the coefficients of a local polynomial expansion of the time-varying parameters In Kenné et al [8], a sliding mode approach is proposed where convergence of the parameter estimates to the true estimates is guaranteed The technique relies on an estimate of the sign of the parameter estimation error which may be very difficult to obtain in practice Many researchers have considered the problem of unknown input estimation which is very closely related to problem of estimation of time-varying parameters Combined state and unknown input estimation for a class of nonlinear systems is proposed in Ha and Trinh [4] High-gain observer methods are proposed in Triki et al [] A sliding-mode observer design approach is proposed in Veluvolu and Soh [] for the estimation of unknown inputs In contrast to the estimation of time-varying parameters, the estimation of unknown inputs rely on often restrictive structural assumptions which are related to the observability of the inputs from the available measurement In parameter estimation, this structural information would in general be limited to an identifiability requirement or a persistency of excitation condition The analysis of nonlinear systems with time-varying parameters has been considered in Bastin and Gevers [988] In this paper, the authors propose an adaptive observer canonical form and propose an adaptive observer or this class of systems It is claimed that the estimation of timevarying parameters can be performed subject to a bound on the rate of change of the parameters A related result is presented in Marino et al [] where an adaptive observer is proposed for a class of nonlinear systems with time-varying parameters and bounded disturbances As in Bastin and Gevers [988], estimation of the time-varying parameters is possible subject to a bound on the derivative of the parameters The guiding principle in the estimation of time-varying systems is that one can estimate parameters up to a bound on the rate of change of the parameters As a result, the IFAC, All rights reserved 39
Furama Riverfront, Singapore, July -3, proposed technique can be overly conservative since it is almost impossible to estimate this bound a priori In this paper, we propose a set-based estimation technique for the estimation of time varying parameters in linearly parameterized nonlinear systems The approach proposed is influenced by the identification scheme presented in Adetola and Guay [] generalizes the approach from earlier Adetola and Guay [9] and Adetola and Guay [] to a class of nonlinear systems with time-varying parameters This method ensures convergence of the parameters to a neighbourhood on their true values and provides an estimate of the size the uncertainty set using a set-update algorithm that guarantees non-exclusion of the true parameter estimates This paper is organized as follows The problem description is given in Section The parameter estimation routine and uncertainty set adaptation are presented in Section 3 This is followed by a simulation example in Section 4 and a brief conclusion in Section 5 PROBLEM DESCRIPTION Consider a nonlinear system p ẋ = fx + g i xθ p t = fx + gxθt i= where fx and g i x i =,, n are [ sufficiently smooth ] vector valued functions, gx = g x,, g p x R n p, x R n is the state, y R r is the output, θt R p is an unknown time varying bounded parameter vector assumed to be uniquely identifiable lying within a known compact set Θ = Bθ, z θ, where θ = θ is a nominal parameter value, z θ is the radius of the parameter uncertainty set The vector-valued functions fx, gx is sufficiently smooth The following assumptions are made about Assumption : The state variables xt X a compact subset of R n Assumption : The system is observable Assumption 3: θt represents the time varying parameter such that θt < c The aim of this work is to provide the true estimates of plant s time varying parameters and estimation of the state 3 PARAMETER AND UNCERTAINTY SET ESTIMATION 3 Parameter estimation Let the estimator model for be chosen as ˆx = fx + gxˆθt + Ke + w T ˆθt, K >, ẇ T = Kw T + gx, wt = 3 resulting in the state prediction error e = x ˆx and an auxiliary variable η = e w T θt dynamics: where et = xt ˆxt, ė =gx θt Ke w T ˆθt 4 η = Kη w T θt, ηt = et 5 An estimate of η is generated from ˆη = K ˆη 6 with resulting estimation error η = η ˆη dynamics η = K η w T θt, ηt = 7 An estimate of η is generated from 6 with resulting estimation error η = η ˆη dynamics given by 7, ˆηt = ηt = et Therefore ηt = Consider a Lyapunov function V η = ηt η, it follows from 7 that [ ] V η = η T K η w T θt = η T K η + η T w T θ η T K η + k ηt ww T η + 8 θt θ k where k is some positive constant Thus if one assigns K as K = k I + k wwt for some positive constant k with I, the identity matrix, the inequality above becomes: V η k V η + k θ k V η + c 9 k Since η = ˆη = e, it follows that η = The inequality 9 yields: or t V η t e kt V η + ηt c k k e kt τ dτ c k c k k Following Adetola [8], the parameter estimation scheme has been generated for the above mentioned system 3 Parameter adaptation Let Σ R n θ n θ be generated from Σ = ww T k T Σ, Σt = αi, where k T is a positive constant to be assigned Based on,3 and 6, one considers the parameter update law as proposed in Adetola and Guay [9] is given by Σ = Σ ww T Σ + k T Σ, Σ t = I, { α ˆθt =proj γσ we ˆη, ˆθt }, ˆθt = θ Θ, where Proj{φ, ˆθt} denotes a Lipschitz projection operator Krstic et al [995] such that Proj{φ, ˆθt} T θt φ T θt, 3 ˆθt Θ = ˆθt Θ, t t 4 where Θ Bˆθt, z θ, where ˆθt and z θ are the parameter estimate and set radius found at the latest set update respectively Assumption There exists constants α > and T > such that t > +T t wτwτ T dτ α I 5 39
Furama Riverfront, Singapore, July -3, Theorem Let Assumption be met by the trajectories of The identifier composed of and the parameter update law is such that the estimation error θt = θt ˆθt is bounded and such that lim θt K t where K > is a strictly positive constant Proof: Consider the Lyapunov function, V θt = θt T Σ θt it follows from that V θt = θt T Σt θt kt θt T Σ θt + θt T ww T θt 6 If ones substitutes for and w T θt = e η ˆη, it follows that, invoking the properties projection algorithm, V θt θt T Σt θt k T θt T Σ θt θt T we ˆη + θt T ww T θt θt T Σt θt k T θt T Σ θt e ˆη η T e ˆη + e ˆη η θt T Σt θt k T θt T Σ θt e ˆηT e ˆη + ηt η By Young s Inequality on the first term, one obtains, V θt k 3 θt T Σt θ + k θt Σt θt 3 k T θt T Σt θt e ˆηT e ˆη + ηt η where k 3 is some positive constant It follows that if one lets k T = k 3 + k 4 for a positive constant k 4 then V θt k 4 θt T Σt θt e ˆηT e ˆη + k θt Σt θt + 3 ηt η 7 Next, we claim the boundedness of the matrix Σt as follows By integration, one gets: Σt =e k T t Σ + t T e k T t τ wτwτ T dτ e k T t τ wτwτ T dτ e k T T α I = γ I By the boundedness of wt, one can also write, Σt Σ + β e k T t τ dτi αi + β I = γ I As a result, we get that: γ I Σt γ I and γ I Σt γ I As a result, 7 yields: V θt k 4 V θ e ˆηT e ˆη + γ c + c 8 k 3 k V θt e k4t V θ + e k4t V θ + e k4t τ γ + k 3 k c γ k 3 + k c dτ Taking the limit as t, one gets lim V θt γ + c t k 3 k or lim θt γ + c t k 3 k γ As a result, the parameter estimation error enters a small neighbourhood of the origin parameterized by k and k 3, two adjustable positive gains 33 Parameter set adaptation The analysis from the last section indicates that it will not be possible in general to guarantee that lim t θt = However, one can always guarantee that the parameter estimation error belongs to a neighbourhood of the origin In this section, we propose an algorithm that yields an estimate of this neighbourhood which we call the uncertainty set The uncertainty set is taken to be a ball of radius z θ centred at the origin Using the analysis from the previous section, an update law that measures the worst-case progress of the parameter identifier in the presence of a disturbance is given by V zθ t z θ t = 9 4λ min Σt V zθt = 4λ max Σ zθ V zθ t = k 4 V zθ t e ˆηT e ˆη + V zη + λ max Σt c k 3 V zη t = k V zη t + c k, V zη = where V zθ t represent the solutions of the ordinary differential equation with the initial condition, and V zη t is the solution of the initial value problem The parameter uncertainty set, defined by the ball Bˆθ c, z c is updated using the parameter update law and the error bound 9 according to the following algorithm: Algorithm Intialize z θ t i = zθ, ˆθt i = ˆθ At time t i, update ˆθ, Θ = ˆθt i, Θt i, if z θti z θ t i c t i+ t i ˆθ i+ ˆθ i ˆθt i, Θt i, otherwise 3 Iterate back to step, incrementing i = i + 393
θ 8th IFAC Symposium on Advanced Control of Chemical Processes Furama Riverfront, Singapore, July -3, Algorithm ensures that Θ is only updated when the value of z θ has decreased by an amount which guarantees a contraction of the set Moreover z θ evolution as given in 9 ensures non-exclusion of θt as given below Lemma The evolution of Θ = B, z θ under, 9 and Algorithm is such that θ Θt = θ Θt t t Θt Θt, t t t Proof: By definition, we can establish that V η t V zη t From9, it follows that V η t k V η t + c k By the comparison lemma, the solution of the differential equation guarantees that V η t V zη t t t Similarly, we know that V θt V zθ t by definition It follows from 7 that V θt k 4 V θt e ˆηT e ˆη + V zη + λ max Σt c k 3 Hence, by the comparison lemma, the solution of the initial value problem, is such that V θt V zθ t t t 3 and since V θt = θt T Σ θt, it follows that θt V zθ t 4λ min Σt = z θt t t 4 Hence, if θt Θt, then θt B, z θ t, t t If Θt i+ Θt i, then sup θt i z θ t i 5 θt i Θt i+ Now, one can write: sup θt i θt i Θt i+ sup θt i + θt i θt i+ θt i+ Θt i+ z θ t i+ + ˆθt i+ ˆθt i + θt i+ θt i z θ t i+ + ˆθt i+ ˆθt i + c t i+ t i z θ t i which contradicts 5 Hence, Θ update guarantees Θt i+ Θt i And Θ is held constant over update intervals τ t i, t i+ 4 Example 4 SIMULATION EXAMPLE To illustrate the effectiveness of the proposed method, we consider the following system ẋ = x ẋ = θ tx θ t arctan9x + u where θ t = 3 cos5πt and + sint t < 4 θ t = + sin4 + sint 4 4 t < 8 + sin4 + sin8 t 8 For simulation purposes, it is assumed that x = and x = 8 and u = sinπt The parameter estimates are plotted along with their true value in Figure The parameter estimates follow the true unknown parameters effectively The uncertainty radius is compared to the norm of the parameter estimation error in Figure As expected, the norm of the parameter estimation error is always less than the uncertainty radius This confirms that the true unknown uncertain parameters are always contained in the uncertainty set 45 4 35 3 5 5 5 3 4 5 6 7 8 9 Fig Plot of the parameter estimates along with the true parameter values z θ 8 6 4 8 6 4 3 4 5 6 7 8 9 Fig Upper bound of the estimation error norm θ and the uncertainty set radius z θ as a function of time 5 CONCLUSION An adaptive estimation technique is proposed for the estimation of time-varying parameters for a class of z θ θ θ Error 394
Furama Riverfront, Singapore, July -3, x, x 5 5 5 5 5 x x disturbances IEEE Trans Automat Contr, 466:967 97, M Triki, M Farza, T Maatoug, M MSaad, Y Koubaa, and B Dahhou Unknown inputs estimation for a class of nonlinear systems Int J Sci Tech Automat Contr Comput Eng, 4:8 9, KC Veluvolu and YC Soh Multiple sliding model observers and unknown input estimations for lipschitz systems Int J Adapt Contr Sig Proc, :3 34, Y Zhu and PR Pagilla Adaptive estimation of timevarying parameters in linearly parametrized systems J Dyn Syst Meas Contr, 8:69 695, 6 3 4 5 6 7 8 9 Fig 3 State trajectories over the interval [,] continuous-time nonlinear system A set-based adaptive estimation is used to estimate the time-varying parameters along with an uncertainty set The proposed method is such that the uncertainty set update is guaranteed to contain the true value of the parameters Unlike existing techniques that rely on the use of polynomial approximations of the time-varying behaviour of the parameters, the proposed technique does require a functional representation of the time-varying behaviour of the parameter estimates The techniques yields guaranteed bounds for the uncertainty on the parameter estimates that can be utilized for the monitoring and robust control of uncertain systems REFERENCES V Adetola Integrated real time optimization and model predictive control under parametric uncertainities PhD thesis, Queen s University, 8 V Adetola and M Guay Robust adaptive mpc for constrained uncertain nonlinear systems Int J Adapt Control Signal Process, V Adetola and M Guay Robust adaptive mpc for systems with exogeneous disturbances In American Control Conference, St Louis, MO, 9 G Bastin and M Gevers Stable adaptive observers for nonlinear time varying systems IEEE TAC, 33:65 658, 988 J Chen, G Zhang, and Z Li On-line parameter estimation for a class of time-varying continuous systems with bounded dusturbances Int J Adapt Contr Sig Proc, 5:8 3, S Dhaliwal and M Guay State estimation and parameter identification of continuous-time nonlinear systems submitted for publication in Automatica, QP Ha and H Trinh State and input simultaneous estimation for a class of nonlinear systems Automatica, 4:779 785, 4 G Kenné, T Ahmed-Ali, F Lamnabhi-Lagarrigue, and A Arzandé Nonlinear systems time-varying parameter estimation: Application to induction motors Electric Power Systems Research, 78:88 888, 8 M Krstic, I Kanellakopoulos, and P Kokotovic Nonlinear and Adaptive Control Design Wiley: Toronto, 995 R Marino, G L Santosuosso, and P Tomei Robust adaptive observers for nonlinear systems with bounded 395