Microsoft Word - Article+Text,+KrasnovDV+UtkinAV+ASSA+(1) Adv Syst Sci Appl 2024; 04; 57-65 Published online at https://ijassa.ipu.ru. Estimating the Unmeasured Output of Nonlinear SISO System under Uncertainty Dmitry V. Krasnov, Anton V. Utkin* V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia Abstract: Nonlinear single-channel control plant with affine unmached external disturbances are considered. Their mathematical model has the so-called triangular (based on the composition of the arguments of the functions in each equation) “input-output” form, which has the following structural properties. If there are no external disturbances, then this form is controllable and observable in terms of output. If external disturbances are smooth, then the control plant model is representable in the canonical “input-output” form with respect to mixed variables (functions of state variables, external disturbances and their derivatives). In the new mixed variable basis, the system remains controllable, observable, and has the same relative degree as the original system. If the output variable is measured, then an observer of mixed variables can be constructed based on the canonical system and, using suitable feedback, ensure the desired behavior of the output variable. In this case, there is no need to detail external disturbances and design individual observers for them, which greatly simplifies the structure of the controller. This paper discusses the problem of implementing the specified synthesis procedure in the case when the output variable is not measured. Motivating examples are given and conditions are formulated under which it is possible to first restore the output variable from the measurement of other state variables invariantly with respect to the action of external disturbances. Methods for estimating external disturbances using observers with piecewise linear corrective actions, which do not require the introduction of dynamic disturbance models, constitute the methodological basis for solving this problem. Using the example of various mathematical models of single-link manipulators, the fulfillment/failure of the conditions necessary to restore an unmeasured output variable is demonstrated. The proposed approach can be extended to multichannel nonlinear control plants without loss of generality. Keywords: nonlinear SISO system, unmatched disturbances, observability, reduced-order state observer, piecewise linear function with saturation, single-link manipulator. 1. INTRODUCTION The problem of constructing the reduced-order state observers is considered for nonlinear minimal-phase systems with one input and one output (SISO), represented in a triangular (by the composition of the function arguments in each equation) “input-output” form. This form is convenient for synthesis and feedback when solving the tracking problem, and for the state observer for estimating unmeasured variables, when only the output (controlled) variable is measured, and all parametric uncertainties and external disturbances are match, i.e. act on the input in the last equation [12, 15, 17–19]. In this paper, we study a case little studied in theory, but common in practice, when the output (controlled) variable of the tracking system for one reason or another cannot be measured [5], the set of sensors is not complete, and the parametric and external disturbances are unmatched. The problem of observing non-measurable (including controlled) variables is considered in a narrow setting, when the identification of unknown parameters and the construction of external disturbance generators are not provided or are not possible. Our goal is to formulate for a triangular “input-output” system the conditions necessary for constructing a physically * Corresponding author: utkin-av@rambler.ru 58 D.V. KRASNOV, A.V. UTKIN Copyright ©0000 ASSA Adv. in Systems Science and Appl. (0000) realizable observer, which depend not only on the structural properties of observability, but also on the channels of action of parametric and external disturbances. The scientific novelty lies in the identification of a practically significant class of SISO systems with unmatched disturbances, where the problem of estimating unmeasured state variables is feasible without additional identification of the existing parametric and external disturbances. The reduced- order state observer of a special structure with piecewise linear control actions provides a solution to this problem [8, 9, 13], if the differential equations of the measured variables in the control plant model do not contain parametric and external disturbances. Then, the observer is constructed as a copy of these equations, where unmeasured state variables are treated as undefined bounded inputs and estimated with the observer's corrective actions. In this case, the uncertainty of differential equations of unmeasured state variables is allowed. The proposed approach, which provides a given estimation accuracy in a finite time, is fundamentally different from the traditional methods for constructing reduced observers based on differential equations of unmeasured variables [3, 4, 6, 7, 14], which in a narrow setting can be implemented only if the entire control plant model is fully defined. The paper is structured as follows. Design options for a state and disturbance observer with piecewise linear correcting actions for a second-order system are discussed in Section 2. The main result is presented in Section 3. The conditions are formulated necessary for constructing a physically realizable observer for estimating the controlled but unmeasured variable of the n-th order SISO system. As an example, the equations of motion of single- link manipulators with a rigid and elastic type of articulation with the shaft of a DC motor are considered. 2. MOTIVATING EXAMPLES Methods for estimating external disturbances in terms of their act on the control plant are the methodological basis for this study. These methods provide the given accuracy of the estimating signals when certain conditions are met [8, 9, 13]. To explain them, we will consider a triangular SISO system “input-output” of the second order ,)(),( ),(),( 22122 12111 butxxfx txxfx       (2.1) ,0,0 ),( 2 211    b x xxf (2.2) where 2T 21 ),( RXxxx  is the state vector, X is an open bounded area of change of state variables in the control process, )(1 tх is controlled output, ),( 21 ххfi are functionally and parametrically defined expressions satisfying the Lipschitz conditions, Ri are external disturbances, Ru is the control (input), which is assumed to be a known function of time, for simplicity b is assumed to be a known constant. Satisfaction of conditions (2.2) means that system (2.1) is controllable and observable with respect to the output in the absence of external disturbances. All internal and external signals in system (2.1) are assumed to be bounded together with their derivatives in the control process, in particular: ,)(,),(,),( ],,0[,)(,)( 2121 XtxFxxfFxxf Tttt iidt d ii iiii     (2.3) where T is regulation time, iiii FF  ,,, are known constants, in this section everywhere 2,1i . ESTIMATING THE UNMEASURED OUTPUT OF NONLINEAR SISO SYSTEM UNDER UNCERTAINTY 59 Copyright ©0000 ASSA. Adv. in Systems Science and Appl. (0000) Let us consider various variants of measurements in system (2.1) and the corresponding conditions for the physical realizability of observers of unmeasured state variables and external disturbances. Example 2.1: If state variables )(),( 21 txtx are measured in system (2.1), then we can obtain estimates of external disturbance s using two first-order autonomous observers: ,),( ,),( 22122 12111 vbuxxfz vxxfz     (2.4) where Rzi  is the state vector, Rv i are corrective actions of the observer. Taking into account (2.1), (2.4), the system with respect to observation errors Rzx iii  takes the following form .2,1,)(  ivt iii  (2.5) Synthesis of observer (2.4) consists in choosing the parameters of piecewise linear corrective actions        ,/1, ,/1),sign( )(sat iiiii iiii iiii llm lm lmv    (2.6) where ,0const, ii lm so as to stabilize the observation errors and their derivatives with a given accuracy .,)()()(,)( 0 Ttttvttt iiiiiii    (2.7) We can obtain inequalities for the choice of observer parameters that provide (2.7) using the second Lyapunov method [8]. Taking into account the measurements in systems (2.4), (2.5), we set the initial values ./1)0(0)0()0()0( iiiii lxz   Then we can provide ii lt /1)(  during the entire control time ],0[ Tt by choosing the amplitudes of corrective actions (2.6) in the form .0)())sign((  iiiiiiiiiii mmm   (2.8) In the linear zone ii l/1 , system (2.5) and its derivatives take the form .)( ,)( iiiii iiiii lmt lmt       The state variables )(),( tt ii   of this virtual system will converge to the given domains (2.7) in a finite time Ttt  00 0: if the conditions 2,1,if0)( ))(()/( ;if0)( ))(()/(     ilm lmtml lm lmtml iiiiiii iiiiiiiiiii iiiiiii iiiiiiiiiii        (2.9) are met. The simultaneous fulfillment of conditions (2.8)–(2.9) 60 D.V. KRASNOV, A.V. UTKIN Copyright ©0000 ASSA Adv. in Systems Science and Appl. (0000) 2,1,,max 1 ,           i m lm i i i i i iii  (2.10) ensures the solution of the formulated estimation problem (2.7). Example 2.2: If in system (2.1)–(2.2) )(1 tx is measured, and ,0)(1 t then, similarly to (2.4), we can reconstruct )(2 tx with a first-order observer based on the first equation of system (2.1) in the form ., ),(sat, 111111 111111 vfzx lmvvz       (2.11) Then, after analyzing virtual systems 1112111 111211111 11211111 ),( ,),(:/1 );sign(),(:/1       lmxxf lmxxfl mxxfl dt d    we obtain inequalities for choosing the parameters of the observer (2.11) similarly to (2.8)– (2.10) ,,max 1 , 1 1 1 1 1 111         FF m lFm  (2.12) providing 11211111 ),()(,)(  vxxftt   , then ,),,()(~)(),( 1111221211 Tttvxhtxtxvxxf  (2.13) where ),( 111 vxh is the solution of the equation 1211 ),( vxxf  with respect to 2x , which exists due to (2.2). Thus, we applied the technique for estimating external disturbances, presented in the first case, to estimate the unmeasured state variable )(2 tx using the reduced observer (2.11), constructed on the basis of the differential equation of the measured variable, which does not depend on the external disturbance. At the same time, we did not explicitly use the second equation of system (2.1), which depends on the disturbance, so identification 0)(2 t is not needed to solve this problem. The main limitation of this approach is that at the design stage it is necessary to obtain the estimates 11, FF required to tune the observer (2.12), taking into account the specific control law and the admissible range of initial values of the state variables. Note that in this case, the standard reduced observer, which is constructed on the basis of the differential equation of an unmeasured variable, is physically unrealizable in a narrow setting, since the second equation of system (2.1) is under the influence of an external uncontrolled disturbance [6, 7]. Note that due to (2.13) we can reconstruct both )(2 tx and 0)(2 t using a full-order observer .~~),~(sat~ ,~),(, 2222222 2212211 zxlmv vbuzxfzvz     (2.14) Taking into account (2.1), (2.14), the system with respect to observation errors Rzx iii  takes the form ESTIMATING THE UNMEASURED OUTPUT OF NONLINEAR SISO SYSTEM UNDER UNCERTAINTY 61 Copyright ©0000 ASSA. Adv. in Systems Science and Appl. (0000) ),~(~),(),( ),(),( 2222122122 112111   vzxfxxf vxxf     (2.15) where for 1tt  the inequalities 222122212 11211111112222 ),(),( ,),(),(~~   Lzxfzxf Lvxhvxhxx    (2.16) are met, 212, LL are well-known Lipschitz constants. Then, under the conditions similar to (2.10), taking into account (2.16), the estimation )()(,)( 2222 tvtztx   is provided in a finite time .: 212 Tttt  The conditions for observability of the controlled variable, regardless of external disturbances, have the following form: if in system (2.1) )(2 tx is measured, and 0)(2 t , ,0/),( 1212  xxxf then we can reconstruct )(1 tx using a first-order observer similar to (2.11) but based on the second equation of system (2.1), and provide 2212 ),( vxxf  ).,()(~)( 22211 хvhtxtx  In addition, we can obtain estimates of both )(1 tx and )(1 t using a full-order observer, similar to (2.14). Example 2.3: If in system (2.1) only )(1 tx is measured and ,0)(1 t then in a narrow setting, i.e. without introducing a dynamic model of )(1 t , it is impossible to solve the problem of separately estimating the variable )(2 tх and external disturbances. Similarly, if only )(2 tx is measured, and 0)(2 t , then it is impossible to solve the problem of estimating separately the variable )(1 tх and external disturbances in a narrow setting. In the next section, the presented results are used to formalize the observability conditions for the output controlled variable of a single-channel arbitrary order system. 3. MAIN RESULT 3.1. General Case of Triangular SISO System Let us consider a minimum-phase nonlinear multidimensional SISO system, represented in a triangular affine form “input-output” ,),()( ;1,1),,...,,,()...,,,( 21121 butxxfx nitxxxxxxfx nnn iiiii        (3.1) where n n RXxxx  T 1 )...,,( is the state vector, Rtx )(1 is controlled output, Ru is the control, )...,,,( 121 ii xxxf are functionally and parametrically defined expressions satisfying the Lipschitz conditions, b is a known constant. All uncertainties of the control plant model and external disturbances are concentrated in expressions i , in some equations of system (3.1) these terms are absent. It is assumed that in the control process all internal and external signals are bounded similarly to (2.3). The fulfillment of the conditions 0,,1,1,0 )...,,( 1 11      bXxni x xxf i ii (3.2) means that system (3.1) is observable with respect to the output in the absence of disturbances, and the output is controllable with respect to the input. In a narrow setting, when the identification of unknown parameters and the construction 62 D.V. KRASNOV, A.V. UTKIN Copyright ©0000 ASSA Adv. in Systems Science and Appl. (0000) of external disturbance generators are not provided or not possible, there are two options for synthesizing feedback that provides tracking of the output variable )(1 tx of a reference signal )(tg . The first variant [9]: if conditions 1,1,,   niCf in ii  and nСtg )( are satisfied, system (3.1) is representable in the canonical basis of mixed variables (functions of state variables, external influences and their derivatives), for the estimation of which it is enough to measure )(1 tx и ),(tg at the same time, it is possible to provide )()(lim 1 tgtx t   using dynamic feedback. The second variant [1]: if the functions )(),( tgti  are not smooth, then when measuring )(...,),(1 txtx n and )(tg , it is possible to provide a given accuracy of the tracking error using static feedback. With an incomplete set of sensors, the possibility of estimating the state variables necessary for feedback synthesis depends on the observability structure of system (3.1) and the absence of disturbances 0)( tj in specific j -th equations. Let the first variant of feedback synthesis be used for system (3.1)–(3.2), but the controlled variable )(1 tx is not measured. Let us formulate the conditions for its observability regardless of disturbances, based on the motivating examples from Section 2. In what follows, a narrow statement of the problem is considered by default, i.e., without additional identification of existing uncertainties. Theorem 3.1: If in system (3.1)–(3.2) there is i -th equation ( ni ...,,2 ) such that: 1. ,0)( ti ,0/)...,,( 111   xxxf ii ;Xx 2. )(1 txi is measured ( 0)(1  txn ); 3. )(txi is measured or observed independently of disturbances (see Example 2.2); 4. )(...,),( 12 txtx i either are not arguments of the function if , or are measured, or are observed regardless of disturbances, then for estimating )(1 tx it is possible to construct a physically realizable reduced observer. Proof The conditions formulated in the lemma are dictated by the triangular form of the considered system (3.1)–(3.2). In the simplest case, when )(txi ( ni ...,,2 ) and all available arguments of the function if are measured (except for )(1 tx ), ,0)( ti then the estimate of )(1 tx can be obtained using the corrective action of the first-order observer constructed on the basis of the i -th equation of system (3.1) similarly to (2.11)–(2.13): )....,,,()(~)( )...,,( ,, ),(sat, 1211 11       iii iii iiiiii iiiiii xxvhtxtx vxxf vfzx lmvvz     (3.3) Note that if ni  , then the first expression (3.3) will take the form buvz nn  , but the following expressions (3.3) will not change. If )(txi and/or the available arguments )(...,),( 12 txtx i of the function if necessary for restoring )(1 tx (3.3) are not measured, then the conditions for their observability are similar to those formulated in the lemma. For example, if in system (3.1)–(3.2) )(2 tх is measured, then under conditions ,0)( tj ,...,,3,2 kj  1 ik , based on the j th equations, we can construct a physically realizable observer for estimating )(...,),( 13 txtx k similarly to (2.14)– ESTIMATING THE UNMEASURED OUTPUT OF NONLINEAR SISO SYSTEM UNDER UNCERTAINTY 63 Copyright ©0000 ASSA. Adv. in Systems Science and Appl. (0000) (2.16). If in system (3.1)–(3.2) )(3 tх is measured, then under conditions ,0)( tj ,...,,3 kj  1 ik , based on the j th equations, we can construct a physically realizable observer for estimating ),(...,),( 14 txtx k if at the same time )(2 tx is measured or is not an argument of functions jf , etc. □ If the conditions of Theorem 3.1 are satisfied and an estimate )(~ 1 tx can be obtained, then for the second variant of the synthesis of the tracking system, the conditions for constructing physically realizable observers for the remaining unmeasured variables with respect to the measured (observed) variables are verified similarly. For example, if ,0)(1 t then there is the possibility of estimating )(2 tx ; if at the same time 0)(2 t , then and )(3 tx etc. 3.2. Application to Single Link Manipulator As an example, let us consider two systems of the form (3.1)–(3.2), which describe the equations of motion of a single-link manipulator [16] with a rigid joint type with a DC motor shaft ubxcxcx txcxcxcx xx    3332323 3232221212 21 ),(sin ,     (3.4) and with an elastic joint type , ,)( , ),(sin)( , 5554545 54544413434 43 12213212 21 ubxaxax xaxaxxax xx txaxxax xx            (3.5) where )(1 tх is the angular position of the pendulum (controlled output), an overline means that this parameter is not exactly known, .0)( t As we can see, the conditions of the Theorem 3.1 are not satisfied in system (3.4). If the controlled variable )(1 tx is not measured, then in a narrow formulation it is impossible to obtain its estimate from measurements of ),(),( 32 txtx this is possible only in the absence (or identification) of uncertainties .,,, 232221 ccc Thus, in system (3.4), the first variant of the tracking system synthesis is realized only when measuring .1x For the second variant of the synthesis of the tracking system, the minimum set of sensors is measurements ),(),( 31 txtx since, based on the first equation of system (3.4), it is possible to construct a reduced observer for estimating ),(2 tx and it is not necessary to identify the existing uncertainties to solve the observation problem. In system (3.5), the conditions of the lemma are satisfied for 4i and measurements ).(),(),( 543 txtxtx Then the corrective action of the first-order observer constructed on the basis of the fourth equation of system (21) will give the estimate ).(1 tx Sensors )(),( 53 txtx are the minimum set for estimating ).(1 tx In this case, estimates of both )(1 tx and )(4 tx can be obtained using a second-order observer constructed on the basis of the third and fourth equations of system (3.5) [10, 11]. Sensors )(),( 53 txtx are the minimum set for the second variant of the synthesis of the tracking system. In this case, the third-order observer is constructed on the basis of the 64 D.V. KRASNOV, A.V. UTKIN Copyright ©0000 ASSA Adv. in Systems Science and Appl. (0000) unperturbed subsystem 21 54544413434 43 ,)( , xx xaxaxxax xx       (3.6) in the form of 11 454513434 343 ,)( , vz vxazxaz vzz       allows one to obtain the estimates ).(),(),( 214 txtxtx If )(),( 31 txtx are measured, then the observer constructed on the basis of subsystem (3.6) in the form of 11 444413434 343 ,)( , vz vzaxxaz vzz       allows one to obtain the estimates ).(),(),( 254 txtxtx An example of constructing a reduced-order observer for a multilink manipulator whose movements are described by matrix equations of the type (3.4) is presented in the work [11]. Simulation modeling of closed systems (3.4), (3.5) with different sets of sensors and with the corresponding reduced observers confirmed their performance [2, 9, 10]. 4. CONCLUSION The control plant model is often presented in the canonical “input-output” form in order to make it convenient to perform mathematical analysis and synthesis of the tracking system. 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