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62
Transfer functions of regular linear systems Part III: Inversions And Duality
 Trans. Amer. Math. Soc
, 2000
"... We study four transformations which lead from one wellposed linear system to another: timeinversion, flowinversion, timeflowinversion and duality. Timeinversion means reversing the direction of time, flowinversion means interchanging inputs with outputs, while timeflowinversion means doing ..."
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Cited by 118 (18 self)
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We study four transformations which lead from one wellposed linear system to another: timeinversion, flowinversion, timeflowinversion and duality. Timeinversion means reversing the direction of time, flowinversion means interchanging inputs with outputs, while timeflowinversion means doing both of the inversions mentioned before. A wellposed linear system is timeinvertible if and only if its operator semigroup extends to a group. The system is flowinvertible if and only if its inputoutput map has a bounded inverse on some (hence, on every) finite time interval [0; ] ( > 0). This is true if and only if the transfer function of has a uniformly bounded inverse on some right halfplane. The system is timeflowinvertible if and only if on some (hence, on every) finite time interval [0; ], the combined operator from the initial state and the input function to the final state and the output function is invertible. This is the case, for example, if the system is conservative, since then is unitary. Timeowinversion can sometimes, but not always, be reduced to a combination of time and flowinversion. We derive a surprising necessary and sucient condition for to be timeflowinvertible: its system operator must have a uniformly bounded inverse on some left halfplane.
LowGain Control of Uncertain Regular Linear Systems
, 1994
"... It is wellknown that closing the loop around an exponentially stable, finitedimensional, linear, timeinvariant plant with square transferfunction matrix G(s) compensated by a controller of the form (k=s)\Gamma 0 , where k 2 Rand \Gamma 0 2 R m\Thetam , will result in an exponentially stable clos ..."
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Cited by 30 (19 self)
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It is wellknown that closing the loop around an exponentially stable, finitedimensional, linear, timeinvariant plant with square transferfunction matrix G(s) compensated by a controller of the form (k=s)\Gamma 0 , where k 2 Rand \Gamma 0 2 R m\Thetam , will result in an exponentially stable closedloop system which achieves tracking of arbitrary constant reference signals, provided that (i) all the eigenvalues of G(0)\Gamma 0 have positive real parts and (ii) the gain parameter k is positive and sufficiently small. In this paper we consider a rather general class of infinitedimensional linear systems, called regular systems, for which convenient representations are known to exist, both in time and in frequency domain. The purpose of the paper is twofold: (i) we extend the above result to the class of exponentially stable regular systems and (ii) we show how the parameters k and \Gamma 0 can be tuned adaptively. The resulting adaptive tracking controllers are not based on syst...
When is a linear system conservative?
, 2003
"... We derive a number of equivalent conditions for a linear system to be energy preserving and hence, in particular, wellposed. Similarly, we derive equivalent conditions for a system to be conservative, which means that both the system and its dual are energy preserving. For systems whose control op ..."
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Cited by 29 (19 self)
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We derive a number of equivalent conditions for a linear system to be energy preserving and hence, in particular, wellposed. Similarly, we derive equivalent conditions for a system to be conservative, which means that both the system and its dual are energy preserving. For systems whose control operator is onetoone and whose observation operator has dense range, the equivalent conditions for being conservative become simpler, and reduce to three algebraic equations.
Conditions for Robustness and Nonrobustness of the Stability of Feedback Systems with Respect to Small Delays in the Feedback Loop
, 1994
"... It has been observed that for many stable feedback control systems, the introduction of arbitrarily small time delays into the loop causes instability. In this paper we present a systematic frequency domain treatment of this phenomenon for distributed parameter systems. We consider the class of all ..."
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Cited by 28 (10 self)
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It has been observed that for many stable feedback control systems, the introduction of arbitrarily small time delays into the loop causes instability. In this paper we present a systematic frequency domain treatment of this phenomenon for distributed parameter systems. We consider the class of all matrixvalued transfer functions which are bounded on some right halfplane and which have a limit at +1 along the real axis. Such transfer functions are called regular. Under the assumption that a regular transfer function is stabilized by unity output feedback, we give sufficient conditions for the robustness and for the nonrobustness of the stability with respect to small time delays in the loop. These conditions are given in terms of the high frequency behavior of the open loop system. Moreover, we discuss robustness of stability with respect to small delays, for feedback systems with dynamic compensators. In particular, we show that if a plant with infinitely many poles in the closed...
Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems. ESAIM: Control, Optimisation and Calculus of Variations, 7, 421–442. Ababacar Diagne received a Graduate Degree in Applied Mathematics and Computer Science
, 2002
"... Abstract. In this paper we study the frequency and time domain behaviour of a heat exchanger network system. The system is governed by hyperbolic partial dierential equations. Both the control operator and the observation operator are unbounded but admissible. Using the theory of symmetric hyperboli ..."
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Cited by 27 (1 self)
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Abstract. In this paper we study the frequency and time domain behaviour of a heat exchanger network system. The system is governed by hyperbolic partial dierential equations. Both the control operator and the observation operator are unbounded but admissible. Using the theory of symmetric hyperbolic systems, we prove exponential stability of the underlying semigroup for the heat exchanger network. Applying the recent theory of wellposed innitedimensional linear systems, we prove that the system is regular and derive various properties of its transfer functions, which are potentially useful for controller design. Our results remain valid for a wide class of processes governed by symmetric hyperbolic systems. Mathematics Subject Classication. 93D09, 93D25, 80A20, 35L50.
Coprime Factorizations And WellPosed Linear Systems
 SIAM Journal on Control and Optimization
, 1998
"... We study the basic notions related to the stabilization of an infinitedimensional wellposed liner system in the sense of Salamon and Weiss. We first introduce an appropriate stabilizability and detectability notion and show that if a system is jointly stabilizable and detectable then its transfer ..."
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Cited by 27 (14 self)
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We study the basic notions related to the stabilization of an infinitedimensional wellposed liner system in the sense of Salamon and Weiss. We first introduce an appropriate stabilizability and detectability notion and show that if a system is jointly stabilizable and detectable then its transfer function has a doubly coprime factorization in H # . The converse is also true: every function with a doubly coprime factorization in H # is the transfer function of a jointly stabilizable and detectable wellposed linear system. We show further that a stabilizable and detectable system is stable if and only if its input/output map is stable. Finally, we construct a dynamic, possibly nonwellposed, stabilizing compensator. The notion of stability that we use is the natural one for the quadratic cost minimization problem, and it does not imply exponential stability.
Quadratic Optimal Control of WellPosed Linear Systems
 SIAM Journal on Control and Optimization
, 1998
"... We study the infinite horizon quadratic cost minimization problem for a wellposed linear system in the sense of Salamon and Weiss. The quadratic cost function that we seek to minimize need not be positive, but it is convex and bounded from below. We assume the system to be jointly stabilizable and ..."
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Cited by 22 (13 self)
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We study the infinite horizon quadratic cost minimization problem for a wellposed linear system in the sense of Salamon and Weiss. The quadratic cost function that we seek to minimize need not be positive, but it is convex and bounded from below. We assume the system to be jointly stabilizable and detectable and give a feedback solution to the cost minimization problem. Moreover, we connect this solution to the computation of either a (J, S)inner or an Snormalized coprime factorization of the transfer function, depending on how the problem is formulated. We apply the general theory to get factorization versions of the bounded and positive real lemmas. In the case where the system is regular it is possible to show that the feedback operator can be expressed in terms of the Riccati operator and that the Riccati operator is a stabilizing selfadjoint solution of an algebraic Riccati equation. This Riccati equation is nonstandard in the sense that the weighting operator in the quadra...
Conservative boundary control systems
, 2006
"... We study continuous time linear dynamical systems of boundary control/observation type, satisfying a Green–Lagrange identity. Particular attention is paid to systems which have a welldefined dynamics both in the forward and the backward time directions. As we change the direction of time we also in ..."
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Cited by 20 (14 self)
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We study continuous time linear dynamical systems of boundary control/observation type, satisfying a Green–Lagrange identity. Particular attention is paid to systems which have a welldefined dynamics both in the forward and the backward time directions. As we change the direction of time we also interchange inputs and outputs. We show that such a boundary control/observation system gives rise to a continuous time Livšic–Brodskiĭ (system) node with strictly unbounded control and observation operators. The converse is also true. We illustrate the theory by a classical example, namely, the wave equation describing the reflecting mirror.
The Effect of Small Delays in the Feedback Loop on the Stability of Neutral Systems
, 1995
"... It is wellknown that exponential stabilization of a neutral system with unstable difference operator is only possible by allowing for control laws containing derivative feedback. We show that closedloop stability of a neutral system with unstable openloop difference operator obtained by applying ..."
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Cited by 18 (1 self)
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It is wellknown that exponential stabilization of a neutral system with unstable difference operator is only possible by allowing for control laws containing derivative feedback. We show that closedloop stability of a neutral system with unstable openloop difference operator obtained by applying a derivative feedback scheme is extremely sensitive to arbitrarily small timedelays in the feedback loop.
Integral control of infinitedimensional systems in the presence of hysteresis: an inputoutput approach
 ESAIM: CONTROL, OPTIMISATION AND CALCULUS OF VARIATIONS
, 2007
"... This paper is concerned with integral control of systems with hysteresis. Using an inputoutput approach, it is shown that application of integral control to the series interconnection of either (a) a hysteretic input nonlinearity, an L2stable, timeinvariant linear system and a nondecreasing gl ..."
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Cited by 15 (11 self)
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This paper is concerned with integral control of systems with hysteresis. Using an inputoutput approach, it is shown that application of integral control to the series interconnection of either (a) a hysteretic input nonlinearity, an L2stable, timeinvariant linear system and a nondecreasing globally Lipschitz static output nonlinearity, or (b) an L2stable, timeinvariant linear system and a hysteretic output nonlinearity, guarantees, under certain assumptions, tracking of constant reference signals, provided the positive integrator gain is smaller than a certain constant determined by a positivity condition in the frequency domain. The inputoutput results are applied in a general statespace setting wherein the linear component of the interconnection is a wellposed infinitedimensional system.