Journal of Science & Technology 101 (2014) 001-006 Optimal State Observer Design Using Gauss-Newton Algorithm in Output Feedback Nmpc Do Thi Tu Anh, Nguyen Doan Phuoc* Hanoi University of Science and Technology No Dai Co Viet Str Ha Noi, Viet Nam Received November 05 2013; accepted April 22 2014 Abstract In order to utilize state feedback controllers in output feedback nonlinear model predictive control (NMPC), appropriate state observers are required sucfi that the system performance will not be affected by ttie presence of the state obsen/er in combination with the state feedback controller Due to the optimality nature of A/MPC, an optimal observer is more eligible for the existing state feedback predictive controller than other observers according to the separation principle This paper presents an optima! observer which can be online connected to a given state feedback predictive controller without affecting the system performance The optimal observer is designed based on the iterative Gauss-Newton optimization algorithm Keywords: State observer, NMPC, Output feedback control I Introduction Model predictive controllers are based on optimization techniques and applied mainly to discrete-lime systems (2) Z)/',K-^,-y(,.) • where y^^^ denotes the output of the predictive model and p^ (-) denotes the function of prediction (1) where x, — x^(k), ,x^{k) independent state variables vector of 7) of the system, I vector of m input signals, y^ = y^{k), ,y,(k) vector of / output signals of the system The optimization problem in model predictive control is solved repeatedly at every time instant whose duration is exactly the sampling penod T of the system input u{t) and output j/(f) Specifically, in order to obtain u,^ = u(KTJ = u{k) from the input and y, =y{kTJ = y{k) from the output at lime mstant t = kT^, fc = 0,1 ., the controller utilizes a predictive model, often constructed from the mathematical model of the system, to determine a fliture control sequence, namely a^ jj^p ,,, ,J.(,^,,_| in a horizon length of M, which minimizes the foilowmg objective function' * Corresponding Author Tei (+844) 3869 2985 Email p]iuoc,nguyendoan@hust edu error at time instant t = {k+tjT^ in the future [1,2] It is widely known that output feedback linear model predictive control (LMPC) has achieved great success m many applications m process industries [3,4] Nonlinear model predictive control (NMPC), however, is currently restricted to state feedback [1] In order to convert a predictive controller from state feedback into dynamic output feedback form, one could thmk of combining the existing state feedback controller with an appropriate state observer If the system performance is preserved by this combination, the observer is said to satisfy the separation principle Although there have been many nonlinear observers with good approximation such as Lipschitz, high gain, or sliding mode observer, [5], none of them has been successfully applied to output feedback NMPC according to separation principle M N observaiwns^; ^ predictions | Fig Principle of combining an observer with the state-feedback predictive controller Journal of Science & Technology 101 (Zuiij uui-uut> In spite of this fact, since the design of the state feedback predictive controller involves the solution of the optimization problem (2), an optimal observer with the same structure of the objective function, if employed with the controller, will not affect the performance of the resulting output feedback control system Therefore, we address in this paper the optimal observer design problem to be used in output feedback NMPC strategy Once the observer and the predictive controller have the objective functions of the same form, we can combine them together with the only objective function upon which the closed-loop system performance can be analyzed Fig, illustrates this idea For the whole receding horizon M along the time axis as depicted in Fig 1, only the subinterval of fc + JV,ft+ M — contains the predicted values of the system input and output The current time is k + N-\ The objective function (2) of the predictive controller is now rewritten as: E P.K„-tt,.) ^ mm (3) and the future optimal control sequence is: In this paper, we assume the availability of the state feedback predictive controller where the objective fimction in (2) is defined as the quadratic cost or the weighted sum of square of prediction errors and control inputs The objective function of the proposed optimal observer is quadratic in estimation errors in order to conform to the quadratic structure of p_ (•), The optimization problem is solved by using the Gauss-Newton method, so it avoids the need to compute the second derivative of a muhivariate fimchon as well as the inverted Hessian matrix as in the Newton-Raphson algorithm [2] Optimal obverver design Consider a discrete-time nonlinear MIMO system descnbed by the state-space model in (1) Assume that the system state x^ is unbounded The observer design problem with observation window N is slated that, every time the window moves along the lime axis by a sampling period T , corresponding to setting the index k := k + 1, one need to find an estimate x^ of the system based on a approximation of the system model {1): *iH,=M>"J (5) and on the input and output measurements: whose first element u^^^ will be applied to the system The remaining subinterval of \k,lt + N — contains the measurements of the system input and output They are used to estimate the system state Jij, denoted as x^, at time instant ( = kT that satisfies the following cntenon: Y^iS^i,-+,'yt^.) -^ •^+, Vt^, « = 0,1, , i V - i (6) within the observation window such that the difference between i , and the actual value x^, observed from the output, is minimized Specifically, from N pairs of consecutive measurements (6) and the model (5), we have: (4) Theoretically, if the running cost q (•) as well as the parameters A', M are selected such that the objective fiinctions (3) and (4) defined on those separated intervals can be coupled into a unique objective fiinction of the form (2) of the state feedback predictive controller, the observer (4) will not make any effects to the performance of the closed-loop system The system performance is "preserved" in the sense that the stability of the composite moving horizon system, comprising a stabilizing state feedback predictive controller and a moving horizon observer, is guaranteed [6] Since the proposed optimal observer is none but a moving horizon observer, it is obvious that the closed-loop system is stable = I x^,u^,a^^^, ,u^^^_j = fX^,M.) (7) where W = {u^, u ^ ^ ^ J and fi,i^k'K) = ^i^ for ^ l=fofo o/,fori>l The error e^ observed from the system output at time instant k +1 then becomes, e = y^^^ —/i(ij^^,Uj^ ) =:h{x^M,^,) (8) Journal of Science & Technology 101 (2014) 001-006 Consequently, the weighted sum of squares of the observation errors for the whole observation window is given by (9) where P = P^ > denotes an arbitrary weighting matrix We can then select this matnx so as to make the form of the (unctions under the sum notation conform to that of functions p^{-) of the state feedback predictive controller in (2) Finally, once the objective function of the observation errors (9) is obtained, the problem of fmding an estimated state £l which is most appropnate for the discrete-time system (1) from its measurements (6) reduces to the problem of solving an unconstrained optimization: = aigrmnQ{xJ (10) We will now solve the optimization problem (10) using Gauss-Newton iterative method Notice that in equation (8), W^,, i = 0,l, ,,, ,N-1 is known from the input measurements, it is hence possible to write h ( £ , , U ^ i ) := ft, (i,)and the objective function (9) can be rewritten as; «(».) = 9{*.)''l'9(%) ^ where rain g(i() = coi \ ( ^ ( ) i -•• '^n,,{^t) P = diag{P, ,P) (") ^nd Aa;/ jfW^ A£j, + + 2jypq(xjsl) AXj - If the number of state variables satisfies n 0, and the observation window N ^3, Fig, shows that time response of the ophmal observer if v{k) has normal distribution over interval [—0.1 , 0.1] (dashed-dotled) is almost the same as the true response (solid) In particular, the time response of the observer if i;(^') = (dashed) and that of the exact system are identical In other words, the optimal observer recovers the exact state in noiseless case Moreover, the time responses of the observer with three different values of the observation window N are shown in Fig, For u(k) = and the sensor noise of normal distribution over [—0,1 , ] , the plots confirm our finding that increasing Af up to yields much improvement on the performance of the observer It was also found, however, in this example that the algorithm fails for A' > since the composite function /,{•) defined as in (7) approaches infinity and hence q{-) is undetermined Therefore, in contrast to the theory that the observation window can be arbitrarily large, the choice of A^ should be taken with care Notice that although the system (16) is uniformly observable as the output depends linearly on the state, and the observation window is finite, i,e,, the assumptions in the theorem in section are not satisfied, the estimates still converge to the actual states of the system This fact is, however, not in contradiction with the stated theorem since the theorem gives only a sufficient condition for the convergence of the observer Further mvestigation into the proposed optimal observer concerns the convergence properly of the iterative algorithm at each sampling instant Specifically, the Gauss-Newton algorithm is compared to the Newton-Raphson one when they are both apphed to the opHmal observer design The detailed description of the Newton-Raphson observer has been presented m [[2,]], Here, we select the terminating conditions for the Newton-Raphson algorithm to satisfy the norm of gradient of the objective function, i,e,, R^Lless than e^ and the maximum number of iterations equal to s^^^ As shown in Fig with ji(fe) = and A^ = 3, the two methods give the same optimal values of the state at almost all simulation sampling instants, except at k = 16 and k = 28 where the estimates obtained by the Newton-Raphson method fail to achieve \^\ < e^ and the returned values are just those at maximum iteration s = s This c be explained by conjecture that the Newton-Raphson procedures at those instants are not properly initialized The effect of amplitude of the sensor noise has also been studied through simulation results (not shown) It was found that, as the amplitude of the sensor noise is large compared to that of the system output, for instance when the output lends to zero, the performance of the observer with respect to A'^ becomes worse, because the output measurements become less reliable for state recovery Fortunately, it is often required for the system output to follow a non-zero reference in model predictive control strategy, and hence, the effect of sensor noise is not vital Fig Time responses of the opUmal state observer with u{k) = 0.8 and N = Journal of Science & Technology 101 (2014) 001-006 Conclusions and future worl( Rererences In this paper, we have presented a synthesis approach of optimal state observer for discrete-time nonlinear systems The performance of the observer is defmed in terms of a finite horizon quadratic function to be minimized at each sampling instant The use of the Gauss-Newion method in the optimal observer algorithm leads to excellent estimation of the system state even if the output is corrupted with sensor noise of sufficiently small amplitude This has been shown in an illustrative numerical example [I.] Findeisen, R and AUgower, F, (2007)' An introduction to nonlinear model predictive control Research report University Stuttgart In general it can be concluded that combining an optimal nonlinear observer with a NMPC strategy may form a successful approach for tackling output feedback model predictive control Therefore, further research on separation principle, i.e,, the performance of the state feedback predictive control can be recovered by the considered optimal observer, is required The fnst step m this research would be to investigate the closed-loop stability of the observerbased NMPC system Relaxed arguments of dynamic programming might lead to some further development in this matter This will be subject of ftiture research [2 ] Tu Anh, D.T va Phii6c, N.D (2013): Thi^t U b^ quan s^t trang thai loi uu cho bQ dteu khten NMPC phan hoi dau To he presented at Vietnamese Conference on Control and Automation VCCA2013, Da Nang, [3,] Tu Anh,_D,T vh Phuoc, ND (2013): Giiii thi^u vS dieu khien dy bao Phan T He tuyen tinh Proceedings of Scientific Conference Faculty of Electronics Engineering, Thai Nguyen University of Technology, pp 129-138 [4,] Wang, L C, (2009): Model predictive control systems design and implementation using MatLab, Springer [5.] Besancon, G, (2007): Nonlinear Observers and Applications, Sponger [6.] Michalska H, and Mayne D.Q (1995): Moving horizon observers and observer-based control, IEEE Transactions on automatic control, vol, 40,, No 6, pp 995-1006 [7,] Golub G.H, and Van Loan CF (1996) Matnx Computations John Hopkins University Press ... concerns the convergence properly of the iterative algorithm at each sampling instant Specifically, the Gauss- Newton algorithm is compared to the Newton- Raphson one when they are both apphed to... description of the Newton- Raphson observer has been presented m [[2,]], Here, we select the terminating conditions for the Newton- Raphson algorithm to satisfy the norm of gradient of the objective... obtained by the Newton- Raphson method fail to achieve \^\ < e^ and the returned values are just those at maximum iteration s = s This c be explained by conjecture that the Newton- Raphson procedures