Proceeings of the 17th Worl Congress The International Feeration of Automatic Control Separation Principle for a Class of Nonlinear Feeback Systems Augmente with Observers A. Shiriaev, R. Johansson A. Robertsson L. Freiovich Department of Applie Physics & Electronics, Umeå University, SE-91 87 Umeå, Sween Department of Engineering Cybernetics, NTNU, NO-7491 Tronheim, Norway Department of Automatic Control, Lun University, PO Box 118, SE-1 Lun, Sween Abstract: The paper suggests conitions for presence of quaratic Lyapunov functions for nonlinear observer base feeback systems with an input nonlinearity in the feeback path. Provie that the system using state feeback satisfies the circle criterion (i.e., when all states can be measure), we show that stability of the extene system with output feeback control from a (full state) Luenberger-type observer may be conclue using the circle criterion. As another result, we state a separation principle for a class of feeback systems with an input nonlinearity. When only local stability results can be state, our metho provies an estimate of the region of attraction. 1. INTRODUCTION The separation principle plays a key role in the esign of output feeback controllers, which instea of a full-state feeback use signals reconstructe by an observer from available measurements (Simon, 1956; Wonham, 1968). Essentially, it states that to achieve stability of the overall system one can esign the controller an the observer separately i.e., a controller is evelope to stabilize the plant as if all states were measure, while an observer is solely esigne to reconstruct the state vector from available measurements. The separation principle is sai to hol if the system resulting from interconnection of the state feeback controller an the observer is stable for all amissible combinations. The main class of ynamical systems that satisfy the separation principle consists of linear systems e.g., (Simon, 1956; Wonham, 1968). In general, however, the separation principle is not vali for nonlinear systems e.g., Example 1 in (Arcak an Kokotovic, 1, p.196). A few extensions of the separation principle to classes of nonlinear systems can be foun in (Atassi an Khalil, 1999), (Arcak an Kokotovic, 1), (Johansson an Robertsson, ), (Arcak, ). In a series of papers, Arcak an Kokotović stuie observerbase nonlinear feeback systems using the circle criterion for observer esign an for robustness analysis. For certain classes of nonlinear systems e.g., output feeback stabilization of systems relate to the Moore-Greitzer compressor moel (1986) observer-base feeback turns out to be ifficult. One instability-prone example of observer-base feeback control is the following ynamical system (Arcak, ) x1 1 x1 + 1 1 u x 5 (1) y x 1 () 1 The work was partly supporte by the Sweish Research Council uner the grants: 5-418, 6-543. Whereas the finite-escape time problems of observer-base output feeback can be trace to violation of the Lipschitz conition, ynamic output feeback stabilization can be accomplishe. As shown by Shiriaev et al. (3), robust stabilization can be accomplishe without resorting to explicit observers. Consier a ynamical output feeback controller of the form z λ 3x 1 + λ 4 z u λ 1 x 1 + λ z+(c 1 x 1 + c 3 z) 5 (3) where λ i, c j are real constants to be efine. With such a controller, the ynamics of the close-loop system are x1 1 λ 1 1 λ z λ 3 λ 4 x1 + 1 w z w (c 1 x 1 + c 3 z) 5 x 5 (4) A key observation is that the nonlinearity w of Eq. (4) an the linear virtual output of the close-loop system (4) v c 1 x 1 + c 3 z satisfies a passivity relationship for any x 1,, z v w (c 1 x 1 + c 3 z)(c 1 x 1 + c 3 z) 5 x 5 (5) Thus, the problem to esign a controller that reners the origin of the system (1) asymptotically stable can be solve (Shiriaev et al. 3). Whereas such ynamic output feeback control of Eq. (4) can be given an observer interpretation, the separation principle is obviously irrelevant. From a control esign point of view, however, the ecomposition of ynamic output feeback into problems of state-feeback control an observer esign is attractive. Thus, the problem how to re-use a calculate statefeeback control in cases without full access to state measurement prompts further research for the separation principle for classes of nonlinear system without restriction to Lipschitzboune nonlinearities. In this paper, we consier ynamic output feeback in the context of the circle criterion with attention to absolute stability an quaratic constraints as tools 978-1-134-789-/8/$. 8 IFAC 6196 1.318/876-5-KR-11.435
17th IFAC Worl Congress (IFAC'8) for analysis an esign. To this purpose, consier a nonlinear system x Ax+Bu, u ψ(z,t), z Input y Cx (6) where a state vector x R n, a control action u R m, an A, B are constant matrices of appropriate imensions. Suppose that a feeback controller is efine as u ψ(z,t), z z(x) (7) where the function ψ satisfies the quaratic constraint ψ T (z,t) ψ(z,t) κx (8) T x(t) 1 ψ(z,t) κ T 1 x(t) κ (9) ψ(z,t) for all time t an x R n, where κ R m n. Molaner an Willems (198) mae a characterization of the conitions for stability with a high gain margin of feeback systems of the structure ẋ Ax+Bu; z Lx; u ψ(z,t) (1) with ψ(, ) enclose in a sector K 1,K. The following proceure was suggeste to fin a state-feeback vector L such that the close-loop system will tolerate any ψ(, ) enclose in a sector K 1, ) follows from Pick a matrix Q Q T > such that (A,Q) is observable; Solve the Riccati equation PA+A T P K 1 PBB T P+ Q (11) for P >. Take L B T P an formulate the Lyapunov function V(x) x T Px to be use for the proof of robust stability. The algorithm provies a robustness result which fulfills an FPR conition i.e., the stability conition will be that of an SPR conition on L(sI A + K 1 BL) 1 B, the state-feeback esign proceure being base on a circle-criterion proof an involving a solution of a Riccati equation. In (Johansson an Robertsson ), the Molaner-Willems proceure was extene to observer-base feeback. Moreover, it was shown that the nominal pole assignment for control an for the observer ynamics can be mae inepenently, a property ientifie with that of the separation principle. In this paper, we will show that the separation principle hols for a class of nonlinear feeback systems escribe by the Lur e problem with input nonlinarities an ynamic output feeback using observer-augmente esign. I m. PROBLEM FORMULATION Prior to formulation of the problem treate in the paper, let us consier the motivating example ẋ1 1 x1 + u Ax+Bu ẋ 3 1 1 (1) y 1 x Cx u sat(z (+sin (t))), z with a time-varying multiplicative isturbance on the input of the control signal which is saturate at level one an furthermore epens on the unmeasure state. The nonlinearity u satisfies the inequality u 3,. (13) At the same time, the sector of linear stability for the linear part of (1) for some ũ κ so that κ 1, + Therefore, one can hope for robust stability of the close loop system (1) only if the secon constraint u > (14) with κ > 1, is vali. This an the saturation in the nonlinearity of (1) imply that the state, where (14) hols, is between ±1. Combining (13) an (14), we get the quaratic constraint (QC) (3 u)(u ) T ( 3) u ( ) 1 x1 x1 u (15) that remains vali provie < 1. Later, this QC is to be use to solve an associate Riccati equation an to ientify the Lyapunov function for any system which satisfies the QC (15) an the linear ynamics of (1). Note, however, that the linear subsystem (1) is unstable an has an input saturation. Hence we can only hope to show local stability results for this example. Consier a Lyapunov function caniate T x1 x1 V P, P P x T > (16) Along the solutions of (1) we have V T x1 ( A x T P+PA ) x 1 T x1 ( A x T P+PA ) x 1 T ( x1 + 3) u ( ) 1 x1 + x1 + x1 u T PBu T PBu (17) where we have use the sector conition (15). Rewriting (17) as V (u h T x 1, T ) (18) T T u x1 x1 + hu hh x T x1 we get the following equations for P, h T ( x1 A x T P+ PA+ 3 T ( x1 PB+ ) P 3 1, h PB+ ) + hh T x1 T x1 u hu (19) To estimate the area of attraction, we coul use the fact that the set V(x 1, ) < c is invariant w.r.t. the close-loop system 6197
17th IFAC Worl Congress (IFAC'8) ynamics (1). To fin the critical value c, we have that the sector conition is satisfie for 1, so that T 3 c max 1 () 1 1 1 Therefore, the estimate area of attraction is T x1 3 x1 3x 1 1 + 1 (1) Unfortunately the suggeste feeback controller in (1) is not irectly applicable because the variable use in the controller is not accessible through measurements, an only the first component x 1 is measure. Therefore, some moification for the controller of Eq. (1) is neee together with some metho for estimating the area of attraction for the close-loop system. This motivates the problem statement given in the next section..1 Problem Formulation This paper eals with robust output-feeback stabilization of a nonlinear system of the form ẋ Ax+Bu, y Cx, z Lx () u ψ(z,t) (3) where x R n is the state, y R k is the vector of measure variables, z R l is the vector of linear combinations of the state use in the feeback controller, u R m is the control signal; the matrices A, B, C, L have appropriate imensions; ψ(, ) is the nonlinearity in the close-loop system. It is assume that A1 The nonlinearity (3) satisfies the QC (κ z u) T (u κ 1 z), z, u, (4) where κ 1, κ are constant m l matrices; A The system (), (3) satisfies the circle criterion base on the QC (4) that ensures asymptotic stability of (), (3). When y only is available to measurement, one can reconstruct the variable z by the Luenberger observer ˆx A ˆx+ Bu+K(y C ˆx), ẑ L ˆx (5) an use the estimate for the feeback control u ψ (ẑ,t) (6) Interconnection of Eqs. (5) an (6) will provie an output feeback controller for system (), an stability becomes a challenge. One can expect that stability of the overall system (), (5) (6) may be epenent on the choice of the observer gain K. The main contribution of this paper is that for any observer gain K, s.t. (A KC) is Hurwitz, the close-loop system (), (5) (6) is absolutely stable, i.e. for any observer gain K, which makes (A KC) asymptotically stable, a new sector conition (i.e., a new quaratic constraint, ifferent from the given one in Eq. (4)) can be foun for the close-loop system (), (5) (6); it is verifie that for this new quaratic constraint all the conitions of the Circle criterion applie to the closeloop system (), (5) (6) still hol. As a irect consequence of this result, one can infer that for any gain K, that makes (A KC) Hurwitz, the close-loop system (), (5) (6) is asymptotically stable. i.e., the feeback controller (3) an the observer gain K coul be chosen separately; for any gain K, that makes (A KC) Hurwitz, one can explicitly fin a Lyapunov function for the close-loop system (), (5) (6) as a solution of the associate Riccati equation; one can verify robust stability of (), (5) (6). Here the robustness is quantifie in terms of the valiity of the frequency conition for the newly foun QC. The paper is organize as follows. Section 3 contains the preliminaries where the formulation of the circle criterion an a brief escription of essential steps involve in its proof are outline. The main result of the paper is given in Sec. 4. In Sec. 4.1 the example from Sec. is consiere for output feeback control. In Sec. 6, conclusions are rawn. 3. PRELIMINARIES To show the asymptotic stability of the system (), (3) base on the circle criterion, consier a Lyapunov function caniate V(x) x T Px, P P T >. (7) Using relation (4), one gets V xt P x x T P Ax+Bu T x T P Ax+Bu + κ z u u κ 1 z T T x Q11 Q 1 x x x u Q T Q 1 Q u u u where Q 11 A T P+ PA L T κ T κ 1 L L T κ T 1 κ L (8) Q 1 PB+L T (κ1 T + κ), T Q I m Question: Uner what conitions is there a P P T > such that Q in (8) is positive efinite? The answer is given by: Theorem 1. (Frequency Theorem (Yakubovich, 1963)). Let the pair (A,B) be stabilizable, then there is a P P T such that Q of Eq. (8) fulfills Q > if an only if there is ε > such that for any vectors x C n, ũ C m relate by jω x A x+bũ, ω R 1 (9) the following inequality hols { } Re (κ L x ũ) (ũ κ 1 L x) < ε ( x + ũ ). (3) If et( jω A) ω R 1, the inequality (3) is equivalent to new QC vali for all ω R 1 {( ) Re κ L( jωi n A) 1 B I m (31) ( )} I m κ 1 L( jωi n A) 1 B < Suppose that there exists a (virtual) feeback ū Rz (3) that satisfies the QC (4), an such that the matrix (A BRL) is Hurwitz. Then the time erivative of V with the feeback (3) reuces to the Lyapunov inequality V ) x ((A T BRL) T P+P(A BRL) x < (33) 6198
17th IFAC Worl Congress (IFAC'8) As (A BRL) is Hurwitz, it implies that the matrix P is positive efinite. Note that existence of the feeback (3) satisfying the QC is an assumption for the proof an shoul not be confuse with the real close-loop feeback with the nonlinearity ψ( ). Note that completing the squares after aing the quaratic form, which is not negative ue to the impose quaratic constraint, to the erivative of a quaratic Lyapunov function is the classical iea, known as the simplest form of application of the S-proceure. Summing up the arguments, one can formulate: Theorem. (Circle Criterion (Yakubovich, 1963)). Consier the system of Eqs. (), (3), where the nonlinearity satisfies the quaratic constraint (4). Suppose that there exists a linear feeback (3) such that it satisfies the quaratic constraint (4) an makes the matrix (A BRL) Hurwitz. If the frequency conition (3), (9) hols, then the nonlinear interconnecte system (), (3) is globally exponentially stable. Remark 1. The given formulation of the circle criterion oes not preten to be general an we refer to the original papers (Yakubovich, 1963; Zames, 1966) an textbooks such as (Khalil, ) an references therein for ifferent extensions. The frequency conition an the circle criterion are shown here to highlight the ieas behin the evelopment in the next section. Note also the result of (Molaner an Willems, 198), where the circle criterion was use for synthesis of state feeback controllers with specifie gain an phase margins. 4. STABILITY OF THE OBSERVER-BASED SYSTEM Consier the case when the variable z of the system () is not available, while only the output variable y is measure. To form the ynamic output feeback controller, form an observer an let its state substitute the true system state in the state-feeback control. Let an observer be given by (5), an the feeback controller be chosen by Eq. (6). By assumption the nonlinearity u ψ(z,t), see (3), satisfies the QC (4), that is (κ z u) T (u κ 1 z), z, u, (34) where z an u are not necessarily solutions of the close-loop system, but rather coul be seen as any vectors of appropriate imensions. From this observation follows that for any choice of the observer gain K, the solutions of the close-loop system (), (5) (6) with u ψ(ẑ,t) satisfy the following new QC (κ ẑ u) T (u κ 1 ẑ), ẑ, u, (35) The next question arises: Given an observer gain K, uner what conitions coul the circle criterion be applie to the extene nonlinear system (), (5), (6) base on the QC (35)? The answer is given in the next statement: Theorem 3. Suppose all assumptions of Theorem hol. Then for any gain K that makes the matrix (A KC) Hurwitz, the close-loop system (), (5), (6) satisfy all conitions of the circle criterion applie to the quaratic constraint (35) i.e., with such choice of the observer gain the close-loop system (), (5), (6) is robustly globally exponentially stable. Proof. The system equation () augmente with the observer (5) looks as follows A xˆx KC A KC xˆx }{{} A aug Choose a Lyapunov function caniate as B + B u (36) }{{} B aug W X T PX, where X x T ˆx T T (37) Its time erivative along the close-loop system solution is W X T T P A aug X + B aug u + κ ẑ u u κ 1 ẑ T X X Q, Q > (38) u u where Q is efine in Table 1, Eq. (39). From the frequency theorem, it can be conclue that there exists a matrix P P T such that Q > if an only if X C n, ũ C m relate by jω X A aug X + B aug ũ, ω R 1 (4) the inequality vali {( ) ( )} Re m n, κ L X ũ ũ m n, κ 1 L X < (41) To simplify the left han sie of (4), one can use the ientities of the next statement Lemma 1. The following equalities hol 1, I n X, I n jωi n A aug Baug ũ (4), I n ( jωi n A) 1 Bũ Proof of Lemma 1 comes from stanar matrix computations an is omitte. Base on this fact one can rewrite the inequality (4) like the inequality (31). Therefore, from the frequency theorem one conclues that there exists a matrix P P T such that W evaluate along any solution of the system (), (5), (6) is negative efinite. To check that P >, consier the linear subsystem (), (5) with the feeback controller ū Rẑ (43) As assume, this feeback satisfies the QC (4), an at the same time the system () is stabilize by (3). It follows from the separation principle for linear time-invariant systems that the close-loop system (), (5) an (44) is asymptotically stable. The time erivative of W with the feeback (44) is of the form W X T P A aug X + B aug u (44) X T n n P A aug B aug X < RL The last relation is again a Lyapunov inequality, an the asymptotic stability of the matrix n n A aug B aug RL implies that P is positive efinite. Let us comment the result: 1). The main step i.e., checking the frequency conition becomes trivial ue to asymptotic unbiaseness of the estimate 6199
17th IFAC Worl Congress (IFAC'8) A T n n augp +PA aug n n n n L T κ T κ 1 L L T κ1 T κ L Q B T augp + PB aug + m n,l T (κ1 T + κ T ) m n,(κ 1 + κ )L T I m (39) Table 1. Expression for Q in Eq. (38) ˆX, which is a basic property of the Luenberger observer. It implies a solvability of the corresponing Riccati equation for the system augmente by the observer. Positiveness of the solution of Riccati equation, in turn, follows from the valiity of the separation principle for linear systems. ). The Lyapunov functions V an W, constructe from the solutions P an P of the corresponing Riccati equations, have no clear connection. The constructive proceure for esign of a gain K an an appropriate Lyapunov function W, base on V, coul be foun in (Johansson an Robertsson, ). 4.1 Example (cont ) Output Feeback Consier observer-base feeback control for the system (1) ẋ1 1 x1 + u Ax+Bu (45) ẋ 3 1 1 with ˆx 1 ˆ 1 ˆx1 K1 + u+ (y Cz) 3 1 ˆx 1 K (A KC)z+Bu+Ky (46) y 1 x Cx u sat(v (+sin (t))), v ˆ where we have a similar local sector conition as before, now w.r.t. u an ˆ T ( ˆx1 ˆx1 ˆ 3) ˆx (47) u ( ) 1 u Consier the Lyapunov function caniate W X T PX, P P T > with X x 1,, ˆx 1, ˆ T. The relate Riccati equation is A T aug P +PA ( ) aug + (48) 3 T + PB aug + PB aug + 4 4 where A A aug KC A KC an the solution B, B aug B 1.774 3.44 17.6894 1.8689.318 P 3.44.6455 3.756 1.4151, λ(p) 1.153 17.6894 3.756 17.314.17.7983 1.8689 1.4151.17 1.1847 37.94 To estimate the region of attraction for X 4 1 we nee to fin the critical value c max for the invariant set of a maximal ellipsoi W X T PX c max (49) which will lie between the two surfaces ˆ ±1. Obviously, such an ellipsoi exists. As in the state feeback case, we have:.8.6.4...4.6.8.9.7.5.3.1.1.3.5.7.9 Fig. 1. Estimate region of attraction (RoA) in the x 1 plane. Yellow ellipsoi: Estimate RoA ( 3 1 + x < 1) for state feeback case. Re ellipsois: Estimate RoA (projecte) for the observer base control with ifferent initial conitions { ˆx 1 (), ˆ ()}. x 1, c max x 1, x, ˆx 1, 1P x 1, x, ˆx 1, 1T (5) with x 1 x ˆx 1 y 1 y 1 y y 1 y 4 y, 3 y P 1 3 (51) 1 y 4 y 4 1 Note that in inequality (5) we can set the initial conitions of the observer ˆx 1 an ˆ ourselves. If the QC is not satisfie globally, but only for x Ω, an open set containing the origin in its interior, say Ω { < 1}), then estimates of the regions of attraction can be calculate from x T Px < c with the biggest c such that Ω contains this set. Finally, the stability omain can be increase using high-gain observers (Atassi an Khalil, 1999). Example Separation Principle an Global Stability Consier observer-base feeback control of a system with the ouble integrator ynamics (Johansson an Robertsson, ) 1 ẋ x+ u (5) 1 1 1 x x x+ u+k(y C x), K, X (53) 1 1 x x y Cx x (54) 1.73 1. u sgn(l x), L 1.731 1., P (55) 1. 1.73 where W(X) X T PX cf. Eq. (37) an L B T P have been calculate base on a feeback transformation with A BL BL B A, B A KC (56) B L T PB P, PA + A T P Q, (57) L T 6
1 3 4 5 6 7 17th IFAC Worl Congress (IFAC'8) W(X(t)) 1 9 8 7 6 5 4 3 1 Lyapunov function Time t Fig.. Lyapunov function trajectories from switching output feeback control of ouble integrator ynamics. with the weighting matrices 1.73 1. 1.73 1. 4. 1.73 5. 8.98 P 1. 1.73 1. 1.731 1.73 1. 17.66 5.8, Q 1.73. 3.464. 5. 3.464 17.6 14.417 1. 1.73 5.8 11.979 8.98. 14.417 6.716 This example emonstrates the separation principle with global asymptotic stability in the context of observer-supporte high-gain feeback (Fig. ). 5. DISCUSSION We have provie a constructive proceure for observers for Lur e interconnections, exploiting that there are solutions to the Yakubovich-Kalman-Popov equation in the form of nonminimal positive real systems. Thus, the very restrictive SPR conitions relevant to observer-base feeback control systems can be significantly relaxe. Important applications of such systems can be foun in nonlinear stability theory, observer esign an esign of feeback stabilization e.g., by means of the Popov criterion. A metho for construction of Lur e-lyapunov functions for systems with observer-base feeback control is given. By construction, the nonlinear ynamic output feeback control accomplishe exhibits SPR properties, yet non-minimal, with asymptotic stability an passivity properties guarantee for the close-loop system. The circle criterion consiere here, coul be seen as a particular example of the so-calle Quaratic Criterion for absolute stability evelope by Yakubovich an others, see (Yakubovich, 1967; Yakubovich, ). As expecte, a ifferent form of quaratic constraint, such as the Popov criterion, may also result in the valiity of the separation principle. The main obstacle in the irect extension of Theorem 3 for the general case of quaratic constraint is checking conition for the so-calle minimal stability of the augmente system, see (Yakubovich, ; Shiriaev, ). Apart from its relevance to observer-base feeback control, we expect that the new metho will have application to hybri system an to high-gain feeback systems controlle by logicbase switching evices. 6. CONCLUSIONS This paper escribes a class of nonlinear control systems, for which the separation principle is vali. It is assume that the system stability is etermine from the circle criterion, while the non-linearity in the system is relate to a non-stationary state feeback controller. If the state vector of the system, use as input to the feeback controller, is not available, then following the stanar proceure, one can exten the system by a full-orer observer, using the observer state as controller input. This leas to the question of stability of the overall system, even if it is known that the observer state converges to the true one. One can expect that stability may epen on the observer gain chosen. The main contribution of the paper states that this is not the case. For any choice of the observer gain that provies convergence of the observer state to the true one, the extene system is globally asymptotically stable, an an explicit form of a quaratic Lyapunov function for the extene system is erive. REFERENCES Arcak M. an P. Kokotović. Nonlinear observers: a circle criterion esign an robustness analysis, Automatica, 37:193 193, 1. Arcak M. A Global Separation Theorem for a New Class of Nonlinear Observers, Proc. 41st IEEE Conf. Decision & Control, Las Vegas, NV, pp. 676 681, Atassi A. N. an H. K. Khalil, A separation principle for the stabilization of a class of nonlinear systems, IEEE Trans. Automatic Control, 44, 167 1687, 1999. Johansson R. an A. Robertsson. 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Some remarks on System analysis via integral quaratic constraints, IEEE Trans. Automatic Control, 45(8):157 153,. Simon H.A. Dynamical Programming uner Uncertainty with Quaratic Criterion Function, Econometrica, 4, 74, 1956. Wonham W.M. On Separation Theorem of Stochastic Control, SIAM J. Control, 6, 1968. Yakubovich V.A. Frequency conitions of absolute stability of control systems having hysteresis nonlinearities, Dokl. Aca. Nauk SSSR, 149(), 1963. Yakubovich V.A. Frequency conitions for absolute stability of control systems with several non-linear or linear nonstationary blocks, Automatika i Telemekhanika, 6:5 9, 1967. (Eng. transl. in Automation an Remote Control, pp. 857 88, 1968) Yakubovich V.A. Necessity in quaratic criterion for absolute stability, Int. J. Robust an Nonlinear Control, 1:899 97,. Zames G. On the input-output stability of time-varying nonlinear systems Part II: Conitions involving circles in the frequency plane an sector nonlinearities, IEEE Trans. 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