Control system advanced methods william s levine 2ed

www.elsolucionario.net www.elsolucionario.net ✐ ✐ ✐ ✐ www.elsolucionario.net www.elsolucionario.net ✐ ✐ ✐ ✐ ✐ ✐ www.elsolucionario.net www.elsolucionario.net ✐ ✐ ✐ ✐ ✐ ✐ www.elsolucionario.net www.elsolucionario.net ✐ ✐ www.elsolucionario.net www.elsolucionario.net ✐ ✐ ✐ ✐ www.elsolucionario.net www.elsolucionario.net ✐ ✐ ✐ ✐ ✐ ✐ www.elsolucionario.net MATLAB® and Simulink® are trademarks of The MathWorks, Inc and are used with permission The MathWorks does not warrant the accuracy of the text or exercises in this book This book’s use or discussion of MATLAB® and Simulink® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLABđ and Simulinkđ software â 2011 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed in the United States of America on acid-free paper 10 International 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index ISBN 978-1-4200-7364-5 Automatic control I Levine, W S II Title III Series TJ212.C685 2011 629.8 dc22 2010026367 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com ✐ ✐ ✐ ✐ ✐ ✐ www.elsolucionario.net Contents Preface to the Second Edition xiii Acknowledgments xv Editorial Board xvii Editor .xix SECTION I Analysis Methods for MIMO Linear Systems Numerical and Computational Issues in Linear Control and System Theory 1-1 A.J Laub, R.V Patel, and P.M Van Dooren Multivariable Poles, Zeros, and Pole-Zero Cancellations 2-1 Joel Douglas and Michael Athans Fundamentals of Linear Time-Varying Systems 3-1 Edward W Kamen Balanced Realizations, Model Order Reduction, and the Hankel Operator 4-1 Jacquelien M.A Scherpen Geometric Theory of Linear Systems 5-1 Fumio Hamano Polynomial and Matrix Fraction Descriptions 6-1 David F Delchamps Robustness Analysis with Real Parametric Uncertainty 7-1 Roberto Tempo and Franco Blanchini www.elsolucionario.net Contributors xxi MIMO Frequency Response Analysis and the Singular Value Decomposition 8-1 Stephen D Patek and Michael Athans Stability Robustness to Unstructured Uncertainty for Linear Time Invariant Systems 9-1 Alan Chao and Michael Athans 10 11 Trade-Offs and Limitations in Feedback Systems 10-1 Douglas P Looze, James S Freudenberg, Julio H Braslavsky, and Richard H Middleton Modeling Deterministic Uncertainty 11-1 Jörg Raisch and Bruce Francis vii ✐ ✐ ✐ ✐ ✐ ✐ www.elsolucionario.net viii Contents SECTION II 12 Kalman Filter and Observers Linear Systems and White Noise 12-1 William S Levine 13 Kalman Filtering 13-1 Michael Athans 14 Riccati Equations and Their Solution 14-1 Vladimír Kuˇcera 15 Observers 15-1 SECTION III 16 17 Design Methods for MIMO LTI Systems Eigenstructure Assignment 16-1 Kenneth M Sobel, Eliezer Y Shapiro, and Albert N Andry, Jr Linear Quadratic Regulator Control 17-1 Leonard Lublin and Michael Athans 18 19 H2 (LQG) and H∞ Control 18-1 Leonard Lublin, Simon Grocott, and Michael Athans Robust Control: Theory, Computation, and Design 19-1 Munther A Dahleh 20 The Structured Singular Value (μ) Framework 20-1 Gary J Balas and Andy Packard 21 Algebraic Design Methods 21-1 Vladimír Kuˇcera 22 Quantitative Feedback Theory (QFT) Technique 22-1 Constantine H Houpis 23 Robust Servomechanism Problem 23-1 www.elsolucionario.net Bernard Friedland Edward J Davison 24 Linear Matrix Inequalities in Control 24-1 Carsten Scherer and Siep Weiland 25 Optimal Control 25-1 Frank L Lewis 26 Decentralized Control 26-1 M.E Sezer and D.D Šiljak 27 Decoupling 27-1 Trevor Williams and Panos J Antsaklis 28 Linear Model Predictive Control in the Process Industries 28-1 Jay H Lee and Manfred Morari ✐ ✐ ✐ ✐ ✐ ✐ www.elsolucionario.net 76-4 Control System Advanced Methods Σ1 T q x Σ2 R R ΣN Λ1 y Λ2 p ΛL desired passivity property, such as Kelly’s primal algorithm: x˙ = ki (Ui (xi ) − qi ), p = h (y ), (76.6) where ki > 0, Ui (·) is the derivative of the utility function Ui (·), and h (·) is a penalty function that grows rapidly as y approaches the link capacity c The network equilibrium resulting from this algorithm approximates the solution of the Kuhn–Tucker optimality conditions for Equation 76.5 and the stability of this equilibrium follows from passivity properties established in [13] by exploiting the monotone decreasing property of Ui (·) and monotone increasing property of h (·) As an illustration of the design flexibility offered by the passivity framework [13] presented new classes of algorithms, including the following extension of the source control algorithm in Equation 76.6 ξ˙ i = Ai ξi + Bi (Ui (xi ) − qi ), (76.7) x˙ i = Ci ξi + Di (Ui (xi ) − qi ), (76.8) where the matrices Ai , Bi , Ci , Di must be selected such that the transfer function Hi (s) = Ci (sI − Ai )−1 Bi + Di is strictly positive-real (SPR) The benefit of increasing the dynamic order of Equation 76.6 is demonstrated in [13] with an example where the SPR filter design adds phase lead to counter time delays A passivity analysis and a dynamic extension was also pursued in [13] for Kelly’s dual algorithm where, in contrast to the primal algorithm (Equation 76.6), the source controller is static and the link controller is dynamic In addition, the passivity approach has made stability proofs and systematic Lyapunov function constructions possible for primal-dual algorithms where both source and link controllers are dynamic As an illustration, for the following algorithm in [16] x˙ i = fi (xi )(Ui (xi ) − qi ), p˙ = g (p )(y − c ), www.elsolucionario.net FIGURE 76.2 The feedback structure arising in decentralized congestion control R is the routing matrix, Σi blocks represent the source algorithms for updating sending rates xi , and Λl blocks represent the link algorithms for updating link prices pl (76.9) (76.10) where fi (·) and g (·) are positive-valued functions, a Lyapunov function constructed from a sum of storage functions is N M xi x − x ∗ p p − p∗ i dx + dp (76.11) V= xi∗ fi (x) p∗ g (p) i=1 =1 Lyapunov functions obtained within the passivity framework have also been instrumental in robust redesigns against disturbances, time delays, and uncooperative users [17,18] 76.2.2 CDMA Power Control Game Another important resource allocation problem for communication networks is uplink transmission power control in code division multiple access (CDMA) systems Increased power levels ensure longer ✐ ✐ ✐ ✐ ✐ ✐ www.elsolucionario.net Passivity Approach to Network Stability Analysis and Distributed Control Synthesis 76-5 transmission distance and higher data transfer rate, but also increase battery consumption and interference to neighboring users A particularly elegant approach to this problem is a game-theoretic formulation [19] where the cost function for the ith mobile is Ji = Pi (pi ) − Ui (γi (p)), i = 1, , N (76.12) Pi (·) is a penalty function for the power level pi and Ui (·) is a utility function for the signal-to-interference ratio (SIR), given by Lhi pi γi (p) = , (76.13) σ + k=i hk pk Ui (γi ) = log(L + γi ) (76.14) and studied a first-order gradient algorithm which, upon algebraic manipulations, is given by the expression ∂ Ji hi p˙ i = −λi = −λi Pi (pi ) + λi (76.15) ∂ pi σ + k hk pk This algorithm is to be implemented by the mobiles with the help of the feedback signal: ui = hi σ2 + (76.16) k hk pk received from the base station The resulting feedback structure of the network is depicted in Figure 76.3 where Σi represents the update law (Equation 76.15) for pi , h denotes the column vector of channel gains, and Equation 76.16 is represented as a function of y := k hk pk in the feedback block The equilibrium p∗ of Equation 76.15 is unique when Pi (·) is strictly convex and coincides with the Nash equilibrium for the game defined by the cost function (Equation 76.12) Global stability of this equilibrium follows from the Passivity Theorem because premultiplication by h and postmultiplication by hT in Figure 76.3 preserve passivity of the Σi blocks in the feedforward path from input ui − ui∗ to output yi − yi∗ Likewise, the function φ(y) is strictly increasing and thus, the feedback block is passive from input y − y ∗ to output φ(y) − φ(y ∗ ) Reference [20] uses this observation as a starting point to develop classes of passive power update laws and base-station algorithms that include Equations 76.15 and 76.16 as special cases It further employs the u Σ1 p Σ2 – www.elsolucionario.net where hi is the channel gain between the ith mobile and the base station, L is the spreading gain and σ2 is the noise variance To develop a distributed power update law, [20] employed the logarithmic utility function ΣN hT h ϕ(y) : = − y + σ2 y FIGURE 76.3 The feedback structure arising from the distributed power update law (Equation 76.15) The Σi blocks represent the update laws p˙ i = −λi Pi (pi ) + λi ui , h denotes the column vector of channel gains, and Equation 76.16 is represented as a function of y := k hk pk in the feedback block ✐ ✐ ✐ ✐ ✐ ✐ www.elsolucionario.net 76-6 Control System Advanced Methods Lyapunov function obtained from the Passivity Theorem to study robustness of power control algorithms against a time-varying channel gain h(t) which is assumed to be constant in the nominal stability analysis A related paper [21] studies a team-optimization approach to power control rather than the gametheoretic formulation discussed above and pursues a passivity-based design 76.3 Distributed Feedback Design for Motion Coordination We represent a group of agents and their communication structure with a graph that consists of N nodes connected by M links The presence of a link between nodes i and j means that agents i and j have access to the relative distance information xi − xj We assign an orientation to each link and recall that the N × M incidence matrix D of the graph is defined as ⎧ ⎨+1 if node i is the positive end of link k dik = −1 if node i is the negative end of link k (76.17) ⎩ otherwise The assignment of orientation is for analysis only, and the particular choice does not change the results Our objective is to develop distributed feedback laws that obey the information structure defined by this graph and that guarantee the following group behaviors: P1: The velocity of each node approaches a common velocity vector v d (t) prescribed for the group; that is, limt→∞ (˙xi − v d (t)) = 0, i = 1, , N P2: If nodes i and j are neighbors connected by link k, then the difference variable N zk := d kx = =1 xi − xj xj − xi if i is the positive end of link k if i is the negative end of link k (76.18) converges to a prescribed set Ak , k = 1, , M Examples of sets Ak include the origin in a rendezvous problem, or a sphere if the positions of agents must maintain a prescribed distance in a formation From Equation 76.18, the concatenated vectors x := [x1T · · · xNT ]T , T T z := [z1T · · · zM ] (76.19) satisfy z = (DT ⊗ I)x, (76.20) www.elsolucionario.net 76.3.1 Passivity-Based Design Procedure for Position Coordination where I is an identity matrix with dimension consistent with that of xi and “⊗” represents the Kronecker product To achieve P1 and P2 [22] presented a two-step design procedure: Step is to design an internal feedback loop for each node that achieves passivity from an external feedback signal uiext , left to be designed in Step 2, to the velocity error yi := x˙ i − v d (t) This design step is depicted with a block diagram ˜ i represents the passive in Figure 76.4, where Hi represents the open-loop dynamic model of agent i and H block obtained from the internal feedback design Step is to design an external feedback law of the form M uiext := − dik ψk (zk ), (76.21) k=1 where zk ’s are the relative distance variables as in Equation 76.18 and the multivariable nonlinearities ψk (·) are to be designed The feedback law (Equation 76.21) is implementable with locally available signals because dik = only for links k that are connected to node i ✐ ✐ ✐ ✐ ✐ ✐ www.elsolucionario.net Passivity Approach to Network Stability Analysis and Distributed Control Synthesis (a) ui v d(t) (b) Hi yi ~ Hi uiext xi 76-7 xi ∫ FIGURE 76.4 The task of the internal feedback design is to render the plant (a) passive from the external signal uiext to the velocity error yi := x˙ i − v d (t) With this internal feedback, the dynamics take the form of (b) where the H˜ i block is passive The combination of the internal and external feedback loops result in the interconnected system in Figure 76.5, where asymptotic stabilization of the set A = (z, y)| y = 0, z ∈ {A1 × · · · × AM } ∩ R(DT ⊗ I) is synonymous to achieving objectives P1–P2 above To accomplish this stabilization task, we exploit the interconnection structure in Figure 76.5 where premultiplication by DT ⊗ I and postmultiplication by its transpose D ⊗ I preserve passivity properties of the feedforward path The input 1N ⊗ v d (t) does not affect the feedback loop in Figure 76.5 because it lies in the null space of DT ⊗ I We design the nonlinearities ψk to be passive when cascaded with an integrator as in Figure 76.5 so that the feedforward path is passive from input y to output −uext and, thus, the feedback loop is stable from the Passivity Theorem To guarantee passivity of the nonlinearity ψk (zk ) preceded by an integrator, we let ψk (zk ) = ∇Pk (zk ), (76.23) where Pk (zk ) is a nonnegative and sufficiently smooth function defined on an open set Gk in which zk is allowed to evolve To steer zk to Ak while keeping it within Gk , we construct the function Pk (zk ) to grow unbounded as zk approaches the boundary of Gk , and let Pk (zk ) and its gradient ∇Pk (zk ) vanish on the set Ak Using this construction and the passivity properties of the feedback and feedforward paths in Figure 76.5, [22] proves asymptotic stability of the set A As an illustration, consider the point mass model xă i = ui , (76.24) where xi ∈ R2 is the position of each mass and ui ∈ R2 is the force input The internal feedback ui = −Ki (˙xi − v d (t)) + v˙ d (t) + uiext , ∫ 1N ⊗ v d(t) z DT ⊗ I (76.25) ψ1 • x• Ki = KiT > www.elsolucionario.net (76.22) z ∫ ψ ψ2 –uext D⊗ I ψM ∫ ~ H1 uext y – ~ HN FIGURE 76.5 A block diagram representation for the interconnection of the subsystems in Figure 76.4 via the external feedback (Equation 76.21) ✐ ✐ ✐ ✐ ✐ ✐ www.elsolucionario.net 76-8 Control System Advanced Methods and the change of variables yi = x˙ i − v d (t) bring Equation 76.24 to the form x˙ i = yi + v d (t), (76.26) y˙ i = −Ki yi + uiext , (76.27) where the yi -subsystem with input uiext plays the role of the passive block H˜ i in Figure 76.4 To create and stabilize an equilateral triangle formation with unit side lengths while avoiding collisions, we let Ak be the unit circle, Gk = R2 − {0}, and let the potential functions be of the form Pk (zk ) = |zk | σk (s)ds, k = 1, 2, 3, (76.28) σk (1) = lim σk (s) = −∞, s→∞ lim σk (s) = −∞ (76.29) s→0 and such that, as |zk | → 0, Pk (zk ) → ∞ Then, the interaction forces ψk (zk ) = ∇Pk (zk ) = σk (|zk |) z k zk = |zk | (76.30) guarantee asymptotic stability of the desired formation In particular, σk (|zk |) creates an attraction force when |zk | > and a repulsion force when |zk | < 76.3.2 From Point Mass to Rigid-Body Models A key advantage of the passivity framework is its ability to address high-order and complex agent dynamics by exploiting their inherent passivity properties As an illustration, we now study a rigid-body model and design a controller that achieves identical orientation and synchronous rotation of the agents This means that the objectives P1 and P2 in Section 76.3.1 above must now be modified as A1: The angular velocity of each agent converges to the group reference ωd (t); that is, lim (i ωi − ωd (t)) = t→∞ 0, i = 1, N, where i ωi denotes the angular velocity of agent i in the ith body frame A2: Each agent achieves the same attitude as its neighbors in the limit; that is, the orientation matrices Ri satisfy lim RiT Rj = I, i, j = 1, , N t→∞ To achieve objectives A1 and A2 we first design an internal feedback loop τi for each agent i = 1, , N that renders the attitude dynamics ˙ i + i ωi × i Ii i ωi = τi Gi : i Ii i ω www.elsolucionario.net where σk : R>0 → R is a continuously differentiable, strictly increasing function such that (76.31) passive from an external input signal τext i left to be designed, to the angular velocity error Δωi := i ωi − ωd (t) (76.32) One such controller is τi = i Ii ω ˙ d + ωd × i Ii i ωi − fi Δωi + τext i , which indeed achieves strict passivity from τext i fi > 0, (76.33) to Δωi in the error dynamics system: ˙ i + Δωi × i Ii i ωi = −fi Δωi + τext G˜i : i Ii Δω i (76.34) Other designs (possibly using dynamic controllers) that achieve passivity from τext i to Δωi may also be employed in this framework This design flexibility is illustrated in the next section with an adaptive redesign ✐ ✐ ✐ ✐ ✐ ✐ www.elsolucionario.net Passivity Approach to Network Stability Analysis and Distributed Control Synthesis 76-9 To achieve objective A2 using only relative attitude information, we parameterize the relative orientation matrix RT Rj if node i is the positive end of link k (76.35) R˜ k := iT Rj Ri if node i is the negative end of link k using one of the standard parameterizations of SO(3), such as the unit quaternion representation qk = T T q0k qvk , where qvk denotes the vector part of the quaternion We then design an external torque feedback of the form p τext qvl − qv , (76.36) i = p∈Ni− where Ni+ denotes the set of links for which node i is the positive end and Ni− is the set of links for which node i is the negative end To synthesize this external feedback signal, agent i obtains its neighbors’ relative attitudes with respect to its own frame, parameterizes them by unit quaternions, and adds up their vector parts The closed-loop system resulting from the internal and external feedback laws (Equations 76.33 and 76.36) is depicted in Figure 76.6, where the Jk blocks represent the quaternion kinematics, which possess passivity properties from the relative angular velocity ω ˜ k to the vector component qvk of the unit quaternion representation [23] To incorporate the rotation matrices between the body frames, we replace the matrix ¯ which consists of the × D ⊗ I in Figure 76.5 with a new 3N × 3M rotational incidence matrix D, sub-blocks: ⎧ if node i is the positive end of link k ⎨−I (76.37) d¯ ik =: (R˜ k )T if node i is the negative end of link k ⎩ otherwise As shown in [24], stability of the closed-loop system in Figure 76.6 follows from the Passivity Theorem ¯ T and postmultiplication by D ¯ preserve passivity of the feedforward because premultiplication by D path, and because the feedback path is passive by the internal feedback design Although the foregoing arguments are based on the unit quaternion representation of SO(3), they have been generalized in [24] to other parameterizations 76.3.3 Adaptive Redesign Thus far we assumed that the reference velocities v d (t) and ωd (t) in objectives P1 and A1 above are available to each agent in the group A more realistic situation is when a leader in the group possesses or autonomously determines this information, while others have access only to the relative distance and relative orientation with respect to their neighbors Can the agents estimate v d (t) and ωd (t) online from www.elsolucionario.net l∈Ni+ J1 1N ⊗ ωd(t) ωB –T D qv J2 ω˜ D¯ JM G˜1 τ ext Δω – G˜N FIGURE 76.6 A block diagram representation for the network resulting from the attitude coordination scheme (Equations 76.33 and 76.36) The concatenated vector ωB consists of the angular velocities of the agents in their own ¯ is as defined in Equation 76.37 body frames and the rotational incidence matrix D ✐ ✐ ✐ ✐ ✐ ✐ www.elsolucionario.net 76-10 Control System Advanced Methods this relative distance and orientation information? We now present an adaptive redesign from [25] that accomplishes this task by relying only on the connectivity of the graph and the passivity properties of the interconnected system This adaptive redesign modifies the internal feedback loop to assign the estimate vˆ i and i ω ˆ i to agent i, instead of the unknown references v d (t) and ωd (t) To develop update laws for vˆ i and i ω ˆ i , we parameterize v d (t) and ωd (t) as v d (t) = φ j (t)θ j , ωd (t) = γ j (t)β j , (76.38) j j where φ j (t) and γ j (t) are scalar base functions known to each agent, and θ j and β j are vectors available j j ˆ i (t) only to the leader Agent i estimates the unknown θ j and β j by θˆ i and βˆ i , and reconstructs vˆ i (t) and i ω from j j vˆ i (t) = φ j (t)θˆ i , i ω ˆ i (t) = γ j (t)βˆ i (76.39) j The update laws proposed in [25] are of the form ˙ θˆ i = Λi (Φ(t) ⊗ I)uiext , ˙ βˆ i = Δi (Γ(t) ⊗ I)τext i , (76.40) where Λi = ΛTi > and Δi = ΔTi > are adaptation gain matrices, Φ(t), Γ(t), θˆ i , βˆ i denote concatej j nations of φ j (t), γ j (t), θˆ i , βˆ i respectively, and uiext and τext i are the external force and torque feedback ˆ ˆ laws The adaptation of θi and βi continues until the group reaches the desired position and orientation configuration, in which case the external force and torque feedback signals employed in Equation 76.40 vanish Reference [25] proves that this adaptive scheme indeed stabilizes the desired configuration by exploiting the passivity of the adaptation algorithm, which is depicted in Figure 76.7 for position coordination as an additional module in the existing passive feedback loop Parameter convergence is established under additional conditions One situation in which parameter convergence is guaranteed is when the reference velocity v d is constant and the graph is connected, in which case the proof in [25] makes use of the Krasovskii–LaSalle Invariance Principle Another situation is when the target sets in P2 are Ak = {0} and ψ1 ∫ x• 1N ⊗ v d(t) z• z ∫ DT ⊗ I ψ ψ2 D⊗ I ψM ∫ www.elsolucionario.net j=1 ~ H1 uext – ~ HN ∧ v – 1N ⊗ v d(t) ~ × I ⊗ Φ (t)T ⊗ I θ Λ1∫ × ΛN ∫ I ⊗ Φ (t) ⊗ I FIGURE 76.7 Block diagram for the adaptive scheme (Equations 76.21, 76.39, and 76.40) for position coordination Λi must be interpreted as zero for the leader θ˜ denotes the concatenation of the vectors θˆ i − θ The adaptive module in the feedback path preserves the passivity properties of the closed-loop system ✐ ✐ ✐ ✐ ✐ ✐ www.elsolucionario.net 76-11 Passivity Approach to Network Stability Analysis and Distributed Control Synthesis Final configuration Adaptation is turned on at t = 10 |z3| Distance (m) Position –y (m) 3.5 Configuration before adaptation turned on 2.5 |z2| 1.5 Trajectory of leader x1 Trajectory of agent x2 0 |z1| 0.5 Trajectory of agent x3 Initial configuration |z1| |z2| |z3| 4.5 0 Position –x (m) 10 15 20 25 30 T(s) FIGURE 76.8 Left: Snapshots of the formation in the adaptive design with constant reference velocity v d The adaptation is turned on at t = 10, after which point the trajectories converge to the desired formation Right: The relative distance variables zk plotted as a function of time the regressor vector Φ(t) is persistently exciting, in which case parameter convergence is established with an application of the Teel–Matrosov Theorem [26] As an illustration of the adaptive redesign, we now revisit the example (Equation 76.24) and suppose that v d (t) is available only to agent We modify the feedback law (Equation 76.25) for the agents i = 2, as ui = −Ki (˙xi − vˆ i ) + v˙ˆ i + uiext , Ki = KiT > 0, (76.41) where the signal vˆ i and its derivative v˙ˆ i are available for implementation from the parametrization (Equation 76.39) and the update law (Equation 76.40) In the simulation presented in Figure 76.8, we take the constant reference velocity v d = [0.2 0.2]T and start with the adaptation turned off Since agents 2, possess incorrect information about v d and since there is no adaptation, the relative distances zk not converge to their prescribed sets Ak = {zk : |zk | = 1} At t = 10, we turn on the adaptation for agents and 3, which results in convergence to the desired distances |zk | = asymptotically As a case study for the adaptive redesign, [27] investigated a gradient climbing problem in which the leader performs extremum seeking to reach the minima or maxima of a field distribution and the other agents maintain a formation with respect to the leader To incorporate an extremum seeking algorithm in the motion, [27] let the reference velocity v d (t) be determined autonomously by the leader, in the form of segments vkd (t), t ∈ [tk , tk+1 ], that are updated in every iteration according to the next Newton step To calculate this Newton step, the leader performs a dither motion from which it collects samples of the field and generates finite-difference approximations for the gradient and the Hessian The adaptive redesign discussed above treats the Newton direction as the unknown parameter vector in Equation 76.38 and allows the followers to estimate this direction This case study raised several new problems which led to a refinement of the basic adaptive procedure Among these problems is a judicious tuning of the design parameters to ensure that the followers respond to the Newton motion while filtering out the dither component www.elsolucionario.net 76.4 Passivity Approach to Biochemical Reaction Networks 76.4.1 The Secant Criterion for Cyclic Networks Reference [28] proposed a passivity-based analysis technique for cyclic biochemical reaction networks, where the end product of a sequence of reactions inhibits the first reaction upstream This technique recovered the secant criterion [7,8] developed earlier by mathematical biologists for the local stability of ✐ ✐ ✐ ✐ ✐ ✐ www.elsolucionario.net 76-12 Control System Advanced Methods H2 H1 Hn – such reactions and strengthened it to become a global stability test Cyclic reaction networks are of great interest because, as surveyed in [29], they are widespread in gene regulation, cell signaling, and metabolic pathways Unlike positive feedback systems which constitute a subclass of monotone systems [30], the negative feedback due to inhibition gives rise to the possibility of attractive periodic orbits Indeed, a Poincaré–Bendixson Theorem proven in [31] for negative feedback cyclic systems of arbitrary order shows that bounded trajectories converge either to fixed points or to periodic orbits Stability criteria for cyclic networks are thus important for determining which parameter regimes guarantee convergence to fixed points and which regimes yield oscillations To evaluate local stability properties of negative feedback cyclic systems, [7,8] analyzed the Jacobian linearization, represented in Figure 76.9 as the feedback interconnection of first-order linear blocks Hi (s) = γi /(τi s + 1) They then proved that the interconnected system is Hurwitz if the dc gains γi satisfy the secant criterion: γ1 · · · γn < sec(π/n)n (76.42) In contrast to a small-gain condition which would restrict the right-hand side of Equation 76.42 to be 1, the secant criterion exploits the phase properties of the feedback loop and allows the gain to be arbitrarily large when n = 2, and to be as high as when n = Local stability of the equilibrium proven in [7,8], however, does not rule out the possibility of periodic orbits as shown in [29] with the example: x˙ = −x1 + ϕ(x3 ), x˙ = −x2 + x1 , x˙ = −x3 + x2 , (76.43) where the nonlinearity ϕ(x3 ) = e−10(x3 −1) + 0.1sat(25(x3 − 1)) has a negative slope of magnitude γ3 = 7.5 at the unique equilibrium x1 = x2 = x3 = With γ1 = γ2 = 1, the local secant criterion (Equation 76.42) guarantees asymptotic stability of the equilibrium as in Figure 76.10a However, the numerical simulation in Figure 76.10b indicates that an attractive periodic orbit exists in addition To develop a global stability test for cyclic networks, [28] exploited a passivity property that is implicit in the local secant criterion [32] To make this property explicit, the first step in [28] is to break down the network into n nonlinear subsystems representing the dynamics of each species, interconnected according to the cyclic structure in Figure 76.5 The second step is to verify, for each block Hi , the output strict passivity (OSP) property [33,34]: S˙ i ≤ −yi2 + γi ui yi , www.elsolucionario.net FIGURE 76.9 Cyclic feedback interconnection of dynamic blocks H1 , , Hn (76.44) where ui and yi denote the input and the output of Hi , and Si (xi − xi∗ ) is a positive definite storage function of the deviation of the concentration xi from its equilibrium value xi∗ The third step is to construct a Lyapunov function for the network from a weighted sum of these storage functions Si and to prove that a set of weights that render its derivative negative definite exists if and only if the secant criterion (Equation 76.42) holds In this new procedure, the first-order blocks Hi (s) = γi /(τi s + 1) employed in the local secant criterion are replaced by nonlinear passive systems with OSP gain γi as in Equation 76.44 and the secant condition (Equation 76.42) guarantees global asymptotic stability for the network How does one verify the OSP property (Equation 76.44) and estimate the gain γi ? Although this task may appear intractable for highly uncertain biological models, [28,35] gave procedures to verify OSP without explicit knowledge of the nonlinearities and the equilibrium value xi∗ Instead, one verifies OSP by qualitative arguments that exploit the monotone increasing or decreasing properties of the nonlinearities, ✐ ✐ ✐ ✐ ✐ ✐ www.elsolucionario.net Passivity Approach to Network Stability Analysis and Distributed Control Synthesis (a) 1.3 76-13 (b) 1.4 1.25 1.3 1.2 1.15 1.2 1.1 1.1 1.05 1 0.95 0.9 0.9 0.8 0.8 0.8 0.9 1.1 1.2 1.3 0.7 0.5 1.5 FIGURE 76.10 The trajectories of Equation 76.43 starting from initial conditions (a) x = [1.1 1.1 1], and (b) x = [1.2 1.2 1.2], projected onto the x1 − x2 plane The trajectory in (a) converges to the equilibrium x1 = x2 = x3 = 1, while the trajectory in (b) converges to a periodic orbit such as Michaelis–Menten and Hill equations that arise in activation and inhibition models in enzyme kinetics [36] Once OSP is verified, an upper bound on the gain γi is obtained by inspecting the maximum slope of the steady-state characteristic curve y¯ i = ki (u¯ i ), where y¯ i denotes the steady-state response of the output yi to a constant input ui = u¯ i As an illustration of the global secant test, consider the following simplified model of a mitogen activated protein kinase (MAPK) cascade with inhibitory feedback, proposed in [37,38]: b x1 d1 (1 − x1 ) μ + c1 + x1 e1 + (1 − x1 ) + kx3 b x2 d2 (1 − x2 ) x˙ = − + x1 c2 + x2 e2 + (1 − x2 ) b x3 d3 (1 − x3 ) x˙ = − + x2 , c3 + x3 e3 + (1 − x3 ) x˙ = − (76.45) (76.46) (76.47) where the variables xi ∈ [0, 1] denote the concentrations of the active forms of the proteins, and the terms − xi correspond to the inactive forms (after an appropriate nondimensionalization that scales the sum of the active and inactive concentrations to 1) The inhibition of x1 by x3 in this model is due to the decreasing function μ/(1 + kx3 ) in Equation 76.45, the steepness of which is determined by the parameter k With the coefficients b1 = e1 = c1 = b2 = 0.1, c2 = e2 = c3 = e3 = 0.01, b3 = 0.5, d1 = d2 = d3 = 1, μ = 0.3, we obtained the OSP gains γi numerically for various values of k This numerical experiment showed that the secant condition γ1 γ2 γ3 < is satisfied in the range k ≤ 4.35, which reduces the gap between the small-gain estimate k ≤ 3.9 given in [39] and the Hopf bifurcation value k = 5.1 The secant test further guarantees global asymptotic stability which cannot be ascertained from a bifurcation analysis www.elsolucionario.net 0.85 76.4.2 Generalization to Other Network Topologies The passivity-based analysis outlined above makes the key property behind the local secant criterion explicit, extends it to be a global stability test, and further opens the door to a generalization of this test to network topologies other than the cyclic structure Reference [35] achieved this generality by representing the reaction network with a directed graph and by making use of the concept of diagonal stability [9,10] to expand the passivity-based analysis to such a graph The nodes x1 , , xn in this directed graph denote the concentrations of the species and the links k = 1, , m represent the reactions, as in Figure 76.11 In ✐ ✐ ✐ ✐ ✐ ✐ www.elsolucionario.net 76-14 Control System Advanced Methods where γl is the OSP gain for lth subsystem as in Equation 76.44 and the sign of a link is +1 or −1 depending on whether it represents positive or negative feedback This matrix incorporates information about the passivity properties of the subsystems, the interconnection structure of the network, and the signs of the interconnection terms To determine the stability of the network, [35] checks the diagonal stability of E; that is, the existence of a diagonal solution P > to the Lyapunov equation E T P + PE < When such a P exists, its diagonal entries serve as the weights of the storage functions in a composite Lyapunov function It is important to note that here diagonal stability is not used as a local stability test, but is employed to construct a composite Lyapunov function as in the large-scale systems literature [2,40,41] By taking into account the signs of the off-diagonal terms in Equation 76.48, however, the diagonal stability test exploits the phase properties of the feedback loops in the network and avoids the conservatism of small-gain-type dominance approaches prevalent in large-scale systems studies This diagonal stability test encompasses the secant criterion because, as shown in [28], when E is constructed according to the cyclic interconnection topology, its diagonal stability is equivalent to the secant condition (Equation 76.42) For other practically important network structures, [35] obtained variants of the secant criterion by investigating when the dissipativity matrix E is diagonally stable As an illustration, for the feedback configurations (a) and (b) in Figure 76.11, the dissipativity matrices obtained according to Equation 76.48 are ⎡ ⎢− γ1 ⎢ ⎢ ⎢ ⎢ Ea = ⎢ ⎢ ⎢ ⎢ ⎣ − 0 γ2 −1 1 − γ3 ⎤ −1 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ 1⎦ − γ4 ⎡ ⎢− γ1 ⎢ ⎢ ⎢ ⎢ Eb = ⎢ ⎢ ⎢ ⎢ ⎣ − 0 γ2 −1 1 − γ3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 1⎦ − γ4 (76.49) As shown in [35], matrix Ea is diagonally stable if γ1 γ2 γ4 < 8, and Eb is diagonally stable if γ1 γ2 γ4 < For the feedback configuration in Figure 76.11c, the dissipativity matrix is ⎡ ⎤ 0 − ⎢ γ1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −1 0 ⎥ − ⎢ ⎥ γ ⎢ ⎥ ⎢ ⎥ − ⎥ (76.50) Ec = ⎢ ⎢ ⎥ γ3 ⎢ ⎥ ⎢ ⎥ − ⎢ ⎥ γ4 ⎢ ⎥ ⎣ 1⎦ 0 − γ5 www.elsolucionario.net particular, solid links represent activation terms which play the role of positive feedback and dashed links represent inhibitory terms which act as negative feedback To derive a stability test that mimics the secant criterion, [35] breaks down the network into m subsystems each associated with a link and forms an m × m dissipativity matrix E of the form ⎧ if k = l ⎨−1/γl (76.48) elk = sign(link k) if source(l) = sink(k) ⎩ otherwise, and a necessary condition for its diagonal stability is γ1 γ2 γ4 + γ4 γ5 < ✐ (76.51) ✐ ✐ ✐ ✐ ✐ www.elsolucionario.net 76-15 Passivity Approach to Network Stability Analysis and Distributed Control Synthesis (a) x1 x2 x3 (b) x1 (c) x2 x3 x2 x1 x3 4 Although the necessary condition (Equation 76.51) does not depend on γ3 , a numerical investigation shows that, unlike the feedback configurations (a) and (b), γ3 is implicated in the diagonal stability for the configuration in Figure 76.11c 76.4.3 Passivity as a Certificate of Robustness to Diffusion Thus far we assumed a well-mixed reaction environment and represented the concentration of each species i = 1, , n with a lumped variable xi The study of spatially distributed reaction models is of interest because the presence of diffusion in the spatial domain can lead to subtle instability mechanisms in an otherwise stable reaction system [43] In contrast, the passivity-based stability tests presented in Sections 76.4.1 and 76.4.2 rule out such mechanisms and guarantee robustness against diffusion References [29,35] studied both reaction–diffusion PDE models and compartmental ODE models where the compartments represent a discrete set of spatial domains as further described below, and proved that the secant condition and its generalization in Section 76.4.2 above guarantee global asymptotic stability of the spatially homogeneous equilibrium This homogeneous behavior is illustrated in Figure 76.12 on the MAPK example (Equations 76.45 through 76.47) where the concentrations xi (t, ξ) are now functions of the spatial variable ξ, and the dynamic equation for each i = 1, 2, is augmented with diffusion terms The structural property that ensures robustness against diffusion is particularly transparent in a comj partmental ODE model in which the state vector X j incorporates the concentrations xi of species i = 1, , n in compartment j = 1, , N To make the structure of this Nnth order ODE explicit we introduce a new, undirected, graph G in which the nodes represent the compartments and the links describe the interconnection of the compartments Denoting by LG the Laplacian matrix for this graph, we obtain the block diagram in Figure 76.13, where the feedforward blocks Σj are copies of the lumped x2 (t, ξ) x1 x3 (t, ξ) –1 30 www.elsolucionario.net FIGURE 76.11 Three feedback configurations proposed in [42] for MAPK networks in PC-12 cells The nodes x1 , x2 , and x3 represent Raf-1, Mek1/2, and Erk1/2, respectively The dashed links indicate negative feedback signals Depending on whether the cells are activated with (a) epidermal or (b) neuronal growth factors, the feedback from Erk1/2 to Raf-1 changes sign (c) An increased connectivity from Raf-1 to Erk1/2 is noted in [42] when neuronal growth factor activation is observed over a longer period 20 t 10 0.5 0 ξ 30 20 t 10 0.5 0 ξ 30 20 t 10 0.5 0 ξ FIGURE 76.12 Solutions of the MAPK model (Equations 76.45 through 76.47) augmented with diffusion terms: ξ represents the spatial coordinate and the solutions converge to the spatially homogeneous equilibrium ✐ ✐ ✐ ✐ ✐ ✐ www.elsolucionario.net 76-16 Control System Advanced Methods Σ1 u Σ2 – y ΣN LG ⊗ In reaction model perturbed by the diffusion input uj : Σj : X˙ j = F(X j ) + uj , yj = CX j , uj , yj , X j ∈ Rn , (76.52) and C is a diagonal matrix whose entries are the diffusion coefficients of the species Stability of the interconnection in Figure 76.13 then follows from the results of [44] on positive operators with repeated monotone nonlinearities, extended to multivariable nonlinearities in [45] If the decoupled system (Equation 76.52) with uj = admits a Lyapunov function V (X j ), then, for the coupled system, the sum of these Lyapunov functions for each compartment satisfies ⎡ ⎤ X N N d ⎢ ⎥ j j j N V (X ) = ∇V (X )F(X ) − [∇V (X ) · · · ∇V (X )](LG ⊗ C) ⎣ ⎦ , (76.53) dt j=1 j=1 N X where the first term on the right-hand side is negative definite The second term is due to the coupling of the compartments and includes the repeated nonlinearity ∇V (·) Because the graph Laplacian matrix LG is doubly hyperdominant with zero excess, it follows from [44,45] that the second term on the right-hand side of Equation 76.53 is nonpositive if ∇V (C −1 ·) is a monotone mapping as defined in [45] Indeed, under mild additional assumptions, the Lyapunov functions V (X j ) constructed in [28,29,35] consist of a j sum of convex storage functions of xi which guarantee the desired monotonicity property, thus proving negative definiteness of Equation 76.53 76.5 Conclusions and Future Topics www.elsolucionario.net FIGURE 76.13 Diffusive coupling between the compartments Σj in Equation 76.52 LG is the Laplacian matrix for the graph G representing the interconnection of the compartments, and the concatenated vectors u and y denote T ]T and y = [y T · · · y T ]T u = [u1T · · · uN N The notion of passivity emerged from energy conservation and dissipation concepts in electrical and mechanical systems [11] and became a fundamental tool for nonlinear system design and analysis [33,34] In this chapter we showed that passivity is a powerful design and analysis approach for several types of networks We have further identified recurrent interconnection structures in these networks, such as the symmetric-coupling structure in Figures 76.2, 76.3, 76.5 through 76.7 and 76.13 and the cyclic structure in Figure 76.9, which are well suited to this passivity approach Verification and assignment of passivity properties were hampered by the unavailability of the network equilibrium to the components in the communication and biological network examples in Sections 76.2 and 76.4 This difficulty was overcome by exploiting monotone increasing or decreasing properties of nonlinearities which led to incremental forms of passivity that not depend on the equilibrium location Likewise, the unavailability of the network reference velocity in the motion coordination study of Section 76.3 was overcome with an adaptive redesign Despite these encouraging results, further 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monotone nonlinearities In Proceedings of the 2003 American Control Conference, pp 1861–1866, Denver, CO, 2003 ✐ www.elsolucionario.net 76-18 ✐ ... www.elsolucionario.net T Pr (T) = for continuous-time systems and for discrete-time systems These equations appear in several design/analysis problems, such as optimal control, optimal filtering, spectral... framework for synthesizing robust controllers for linear systems The controllers are robust, because they achieve desired system performance despite a significant amount of uncertainty in the system In... linear system, so that the resulting closed-loop system has a desired set of poles, can be considered an inverse eigenvalue problem The state feedback pole assignment problem is as follows: Given