Constraints can be applied to analyze systems with hysteresis, in spite of the fact that hysteres... more Constraints can be applied to analyze systems with hysteresis, in spite of the fact that hysteresis operators are unbounded and that not all system variable can be expected to approach zero.
It is convenient to think of continuous time (CT) signals as real vector-valued functions of time... more It is convenient to think of continuous time (CT) signals as real vector-valued functions of time t ≤ [0, →), integrable over any bounded interval. From this viewpoint, f1(t) = t (defined, for the sake of mathematical accuracy, as zero at t = 0) and f2(t) = e 2 are signals, while f3(t) = t (where f3(0) = 0) and f4(t) = �(t) (Dirac delta) are not. The set of all signals with values in R will be denoted by L . A continuous time system S with a k-dimensional input and m-dimensional output is simply a map S : L ∞� L (usually multi-valued, so that one input f ≤ L corresponds
A new set of Integral Quadratic Constraints (IQC) is derived for a class of "rate limiters&q... more A new set of Integral Quadratic Constraints (IQC) is derived for a class of "rate limiters", modelled as a series connections of saturationlike memoryless nonlinearities followed by integrators. The result, when used within the standard IQC framework, is expected to be widely useful in nonlinear system analysis. For example, it enables "discrimination" between "saturation-like" and "deadzone-like" nonlinearities and can be used to prove stability of systems with saturation in cases when replacing the saturation block by another memoryless nonlinearity with equivalent slope restrictions makes the whole system unstable. In particular, it is shown that the L 2 gain of a unity feedback system with a rate limiter in the forward loop cannot exceed In addition, a new, more flexible version of the general IQC analysis framework is presented, which relaxes the homotopy and boundedness conditions, and is more aligned with the language of the emerging IQ...
Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207), 1998
A feedback interconnection of neutrally stable linear time-invariant system and a nonlinearity wi... more A feedback interconnection of neutrally stable linear time-invariant system and a nonlinearity with 0 < x¢(x) < kx 2 is called critical since the worst case linearization is at best neutrally stable. This makes the stability analysis of such systems particularly hard. It will be shown that an integrator and a sector bounded nonlinearity can be encapsulated in a bounded operator that satisfies several useful integral quadratic constraints. This gives powerful tools for stability analysis of critically stable systems.
Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171)
In this paper, we design a controller for a class of underactuated nonlinear systems. First, we t... more In this paper, we design a controller for a class of underactuated nonlinear systems. First, we try to nd an appropriate global change of coordinates to transform the dynamics of the system into a desired form which consists of a lower order nonlinear subsystem plus a chain of integrators. Then, we nd a control Lyapunov function (CLF) and its associated control law for the lower order subsystem. Finally, using backstepping procedure we derive the control Lyapunov function and the controller for the whole system. The obtained controller renders the origin semiglobally asymptotically stable. As an special case, we demonstrate this procedure for The Acrobot example which is a twolink planar robot with a single actuator at the elbow.
This paper proposes a model reduction algorithm for discrete-time, markov jump linear systems. Th... more This paper proposes a model reduction algorithm for discrete-time, markov jump linear systems. The main point of the reduction method is the formulation of two generalized dissipation inequalities that in conjunction with a suitably defined storage function enable the derivation of reduced order models that come with a provable a priori upper bound on the stochastic L2 gain of the approximation error. Consider the discrete-time Markov jump linear system (MJLS) G that has the following state space realization: where the state variable is N} and the output is y k ∈ R p . Reduced order model candidates are denoted by Ĝ and it is required that they lie in the same class of MJLS systems, having the state space realization where xk ∈ R n and n < n. In order to quantify the fidelity of Ĝ, an error system E is introduced, whose inputs are the common inputs f k , θ k of G and Ĝ and whose output is the difference of their outputs, namely e k = y kŷk . The parametric input θ k represents the state of a Markov chain that takes values in a finite set Θ = {1, . . . , N} . The transition probability matrix of the Markov chain is denoted by Q and When needed, random variables are denoted by capital letters in order to avoid confusion. The input sequence {f k } is taken to be deterministic and the sequence {Θ k } has the property that ∀k ∈ Z + each Θ k is independent of the state history {X 0 , . . . , X k } up to that point.
Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301), 2002
Nonlinear dynamical systems described by a class of higher index differential-algebraic equations... more Nonlinear dynamical systems described by a class of higher index differential-algebraic equations (DAE) are considered. A quantitative and qualitative analysis of their nature and of the stability properties of their solution is presented. Using tools from geometric control theory, higher index differential-algebraic systems are shown to be inherently unstable about their solution manifold. A qualitative geometric interpretation is given, and the consequences of this instability are discussed, in particular as they relate to the numerical solution of these systems. The paper concludes with ideas and directions for future research.
This paper considers systems over finite alphabets, that is discrete-time systems whose input and... more This paper considers systems over finite alphabets, that is discrete-time systems whose input and output signals take their values in finite sets. Three notions of input/output stability are proposed: gain stability, incremental stability and external stability. Sufficient conditions for stability of feedback interconnections of stable systems, in the finite gain and incremental sense, are derived. Simple examples are used to illustrate the practical significance of these notions of stability as performance objectives and as robustness measures.
In previous works, the authors introduced a framework for software analysis, which is based on op... more In previous works, the authors introduced a framework for software analysis, which is based on optimization of Lyapunov invariants. These invariants prove critical software properties such as absence of overflow and termination in finite time. In this paper, graph models of software are introduced and the software analysis framework is further developed and extended on graph models. A distributed Lyapunov function is assigned to the software by assigning a Lyapunov function to every node on its graph model. The global decremental condition is then enforced by requiring that the Lyapunov functions on each node decrease as transitions take place along the arcs. The concept of graph reduction and optimality of graphs for Lyapunov analysis is briefly discussed.
Proceedings of the 2005, American Control Conference, 2005.
This paper concerns analysis of realtime, safety-critical, embedded software. Software analysis i... more This paper concerns analysis of realtime, safety-critical, embedded software. Software analysis is expected to verify whether the computer code will execute safely, free of run-time errors. The main properties to be analyzed to prove or disprove safe execution include boundedness of all variables and termination of the program in finite-time. Herein the concepts of Lyapunov invariance and associated computational procedures are brought within the context of software analysis. Dynamical system representations of software systems along with specific models that are suitable for analysis via Lyapunov-like functions are developed. General forms for the Lyapunov-like invariants are then constructed in a way to certify the desired properties. Convex optimization methods such as linear programming and/or semidefinite programming are then employed for finding appropriate functions that fit into these general forms and therefore, automatically establish the key properties of software.
2010 Design, Automation & Test in Europe Conference & Exhibition (DATE 2010), 2010
In this paper we present a passive reduced order modeling algorithm for linear multiport intercon... more In this paper we present a passive reduced order modeling algorithm for linear multiport interconnect structures. The proposed technique uses rational fitting via semidefinite programming to identify a passive transfer matrix from given frequency domain data samples. Numerical results are presented for a power distribution grid and an array of inductors, and the proposed approach is compared to two existing rational fitting techniques.
Transactions of the American Mathematical Society, 2004
In a seminal paper, Sarason generalized some classical interpolation problems for H ∞ H^\infty fu... more In a seminal paper, Sarason generalized some classical interpolation problems for H ∞ H^\infty functions on the unit disc to problems concerning lifting onto H 2 H^2 of an operator T T that is defined on K = H 2 ⊖ ϕ H 2 \mathcal {K} =H^2\ominus \phi H^2 ( ϕ \phi is an inner function) and commutes with the (compressed) shift S S . In particular, he showed that interpolants (i.e., f ∈ H ∞ f\in H^\infty such that f ( S ) = T f(S)=T ) having norm equal to ‖ T ‖ \|T\| exist, and that in certain cases such an f f is unique and can be expressed as a fraction f = b / a f=b/a with a , b ∈ K a,b\in \mathcal {K} . In this paper, we study interpolants that are such fractions of K \mathcal {K} functions and are bounded in norm by 1 1 (assuming that ‖ T ‖ > 1 \|T\|>1 , in which case they always exist). We parameterize the collection of all such pairs ( a , b ) ∈ K × K (a,b)\in \mathcal {K}\times \mathcal {K} and show that each interpolant of this type can be determined as the unique minimum...
The problem of stabilizing a second order SISO LTI system of the form ẋ = Ax + Bu, y = Cx with fe... more The problem of stabilizing a second order SISO LTI system of the form ẋ = Ax + Bu, y = Cx with feedback of the form u(x) = v(x)Cx is considered, where v(x) is realvalued and has domain which is all of R 2 . It is shown that, when stabilization is possible, v(x) can be chosen to take on no more than two values throughout the entire state space (i.e., v(x) ∈ {v 1, v2} for all x and for some v1, v2), and an algorithm for finding a specific choice of v(x) is presented. It is also shown that the classical root locus of the corresponding transfer function C(sI -A) -1 B has a strong connection to this stabilization problem, and its utility is demonstrated through several design examples.
The motivation for this paper stems from the need to develop a uniform framework for addressing p... more The motivation for this paper stems from the need to develop a uniform framework for addressing problems in identiÿcation and robust control. System identiÿcation for inÿnite-dimensional Hilbert spaces has been addressed earlier by the authors. System identiÿcation set in an Hilbert space results in uncertain models where the description of non-parametric error is typically a ball belonging to the Hilbert space. The scope of this paper is to complement these results -develop robust control-synthesis and analysis results -for some special, yet, important cases. In this paper we derive a convex parameterization of robustly stabilizing controllers for LTI discrete-time systems deÿned on Hilbert spaces. The perturbations are of rank-one type having both real-parametric and non-parametric components. The parameterization allows for imposing other constraints to obtain meaningful performance from the controller. Analysis tools are also developed for robust stability under SISO block-diagonally structured perturbations. The robustness analysis problem reduces to a ÿnite-dimensional LMI veriÿcation which makes the procedure extremely e cient.
A new method, Fourier model reduction, for obtaining stable, accurate, low-order models of very l... more A new method, Fourier model reduction, for obtaining stable, accurate, low-order models of very large linear systems is presented. The technique draws on traditional control and dynamical system concepts and utilizes them in a way which is computationally very efficient. Discretetime Fourier coefficients of the large system are calculated and used to construct a reduced-order model that preserves stability properties of the original system. Many coefficients can be calculated, which results in a very accurate representation of the system dynamics, but only a single factorization of the large system is required. The resulting system can be further reduced using explicit formulae for balanced truncation. The method is applied to two computational fluid dynamic systems, which model unsteady motion of a two-dimensional subsonic airfoil and unsteady flow in a supersonic diffuser. In both cases, the new method is found to work extremely well. Results are compared to models developed using the proper orthogonal decomposition and Arnoldi method. In comparison with these widely used techniques, the new method is computationally more efficient, preserves the stability of the original system, uses both input and output information, and, for smooth transfer functions, is valid over a wide range of frequencies.
We develop a sufficient condition for the least-squares measurement (LSM), or the squareroot meas... more We develop a sufficient condition for the least-squares measurement (LSM), or the squareroot measurement, to minimize the probability of a detection error when distinguishing between a collection of mixed quantum states. Using this condition we derive the optimal measurement for state sets with a broad class of symmetries. We first consider geometrically uniform (GU) state sets with a possibly nonabelian generating group, and show that if the generator satisfies a certain constraint, then the LSM is optimal. In particular, for pure-state GU ensembles the LSM is shown to be optimal. For arbitrary GU state sets we show that the optimal measurement operators are GU with generator that can be computed very efficiently in polynomial time, within any desired accuracy. We then consider compound GU (CGU) state sets which consist of subsets that are GU. When the generators satisfy a certain constraint, the LSM is again optimal. For arbitrary CGU state sets the optimal measurement operators are shown to be CGU with generators that can be computed efficiently in polynomial time.
IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 2010
This paper presents a system identification technique for generating stable compact models of typ... more This paper presents a system identification technique for generating stable compact models of typical analog circuit blocks in radio frequency systems. The identification procedure is based on minimizing the model error over a given training data set subject to an incremental stability constraint, which is formulated as a semidefinite optimization problem. Numerical results are presented for several analog circuits, including a distributed power amplifier, as well as a MEM device. It is also shown that our dynamical models can accurately predict important circuit performance metrics, and may thus, be useful for design optimization of analog systems.
Constraints can be applied to analyze systems with hysteresis, in spite of the fact that hysteres... more Constraints can be applied to analyze systems with hysteresis, in spite of the fact that hysteresis operators are unbounded and that not all system variable can be expected to approach zero.
It is convenient to think of continuous time (CT) signals as real vector-valued functions of time... more It is convenient to think of continuous time (CT) signals as real vector-valued functions of time t ≤ [0, →), integrable over any bounded interval. From this viewpoint, f1(t) = t (defined, for the sake of mathematical accuracy, as zero at t = 0) and f2(t) = e 2 are signals, while f3(t) = t (where f3(0) = 0) and f4(t) = �(t) (Dirac delta) are not. The set of all signals with values in R will be denoted by L . A continuous time system S with a k-dimensional input and m-dimensional output is simply a map S : L ∞� L (usually multi-valued, so that one input f ≤ L corresponds
A new set of Integral Quadratic Constraints (IQC) is derived for a class of "rate limiters&q... more A new set of Integral Quadratic Constraints (IQC) is derived for a class of "rate limiters", modelled as a series connections of saturationlike memoryless nonlinearities followed by integrators. The result, when used within the standard IQC framework, is expected to be widely useful in nonlinear system analysis. For example, it enables "discrimination" between "saturation-like" and "deadzone-like" nonlinearities and can be used to prove stability of systems with saturation in cases when replacing the saturation block by another memoryless nonlinearity with equivalent slope restrictions makes the whole system unstable. In particular, it is shown that the L 2 gain of a unity feedback system with a rate limiter in the forward loop cannot exceed In addition, a new, more flexible version of the general IQC analysis framework is presented, which relaxes the homotopy and boundedness conditions, and is more aligned with the language of the emerging IQ...
Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207), 1998
A feedback interconnection of neutrally stable linear time-invariant system and a nonlinearity wi... more A feedback interconnection of neutrally stable linear time-invariant system and a nonlinearity with 0 < x¢(x) < kx 2 is called critical since the worst case linearization is at best neutrally stable. This makes the stability analysis of such systems particularly hard. It will be shown that an integrator and a sector bounded nonlinearity can be encapsulated in a bounded operator that satisfies several useful integral quadratic constraints. This gives powerful tools for stability analysis of critically stable systems.
Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171)
In this paper, we design a controller for a class of underactuated nonlinear systems. First, we t... more In this paper, we design a controller for a class of underactuated nonlinear systems. First, we try to nd an appropriate global change of coordinates to transform the dynamics of the system into a desired form which consists of a lower order nonlinear subsystem plus a chain of integrators. Then, we nd a control Lyapunov function (CLF) and its associated control law for the lower order subsystem. Finally, using backstepping procedure we derive the control Lyapunov function and the controller for the whole system. The obtained controller renders the origin semiglobally asymptotically stable. As an special case, we demonstrate this procedure for The Acrobot example which is a twolink planar robot with a single actuator at the elbow.
This paper proposes a model reduction algorithm for discrete-time, markov jump linear systems. Th... more This paper proposes a model reduction algorithm for discrete-time, markov jump linear systems. The main point of the reduction method is the formulation of two generalized dissipation inequalities that in conjunction with a suitably defined storage function enable the derivation of reduced order models that come with a provable a priori upper bound on the stochastic L2 gain of the approximation error. Consider the discrete-time Markov jump linear system (MJLS) G that has the following state space realization: where the state variable is N} and the output is y k ∈ R p . Reduced order model candidates are denoted by Ĝ and it is required that they lie in the same class of MJLS systems, having the state space realization where xk ∈ R n and n < n. In order to quantify the fidelity of Ĝ, an error system E is introduced, whose inputs are the common inputs f k , θ k of G and Ĝ and whose output is the difference of their outputs, namely e k = y kŷk . The parametric input θ k represents the state of a Markov chain that takes values in a finite set Θ = {1, . . . , N} . The transition probability matrix of the Markov chain is denoted by Q and When needed, random variables are denoted by capital letters in order to avoid confusion. The input sequence {f k } is taken to be deterministic and the sequence {Θ k } has the property that ∀k ∈ Z + each Θ k is independent of the state history {X 0 , . . . , X k } up to that point.
Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301), 2002
Nonlinear dynamical systems described by a class of higher index differential-algebraic equations... more Nonlinear dynamical systems described by a class of higher index differential-algebraic equations (DAE) are considered. A quantitative and qualitative analysis of their nature and of the stability properties of their solution is presented. Using tools from geometric control theory, higher index differential-algebraic systems are shown to be inherently unstable about their solution manifold. A qualitative geometric interpretation is given, and the consequences of this instability are discussed, in particular as they relate to the numerical solution of these systems. The paper concludes with ideas and directions for future research.
This paper considers systems over finite alphabets, that is discrete-time systems whose input and... more This paper considers systems over finite alphabets, that is discrete-time systems whose input and output signals take their values in finite sets. Three notions of input/output stability are proposed: gain stability, incremental stability and external stability. Sufficient conditions for stability of feedback interconnections of stable systems, in the finite gain and incremental sense, are derived. Simple examples are used to illustrate the practical significance of these notions of stability as performance objectives and as robustness measures.
In previous works, the authors introduced a framework for software analysis, which is based on op... more In previous works, the authors introduced a framework for software analysis, which is based on optimization of Lyapunov invariants. These invariants prove critical software properties such as absence of overflow and termination in finite time. In this paper, graph models of software are introduced and the software analysis framework is further developed and extended on graph models. A distributed Lyapunov function is assigned to the software by assigning a Lyapunov function to every node on its graph model. The global decremental condition is then enforced by requiring that the Lyapunov functions on each node decrease as transitions take place along the arcs. The concept of graph reduction and optimality of graphs for Lyapunov analysis is briefly discussed.
Proceedings of the 2005, American Control Conference, 2005.
This paper concerns analysis of realtime, safety-critical, embedded software. Software analysis i... more This paper concerns analysis of realtime, safety-critical, embedded software. Software analysis is expected to verify whether the computer code will execute safely, free of run-time errors. The main properties to be analyzed to prove or disprove safe execution include boundedness of all variables and termination of the program in finite-time. Herein the concepts of Lyapunov invariance and associated computational procedures are brought within the context of software analysis. Dynamical system representations of software systems along with specific models that are suitable for analysis via Lyapunov-like functions are developed. General forms for the Lyapunov-like invariants are then constructed in a way to certify the desired properties. Convex optimization methods such as linear programming and/or semidefinite programming are then employed for finding appropriate functions that fit into these general forms and therefore, automatically establish the key properties of software.
2010 Design, Automation & Test in Europe Conference & Exhibition (DATE 2010), 2010
In this paper we present a passive reduced order modeling algorithm for linear multiport intercon... more In this paper we present a passive reduced order modeling algorithm for linear multiport interconnect structures. The proposed technique uses rational fitting via semidefinite programming to identify a passive transfer matrix from given frequency domain data samples. Numerical results are presented for a power distribution grid and an array of inductors, and the proposed approach is compared to two existing rational fitting techniques.
Transactions of the American Mathematical Society, 2004
In a seminal paper, Sarason generalized some classical interpolation problems for H ∞ H^\infty fu... more In a seminal paper, Sarason generalized some classical interpolation problems for H ∞ H^\infty functions on the unit disc to problems concerning lifting onto H 2 H^2 of an operator T T that is defined on K = H 2 ⊖ ϕ H 2 \mathcal {K} =H^2\ominus \phi H^2 ( ϕ \phi is an inner function) and commutes with the (compressed) shift S S . In particular, he showed that interpolants (i.e., f ∈ H ∞ f\in H^\infty such that f ( S ) = T f(S)=T ) having norm equal to ‖ T ‖ \|T\| exist, and that in certain cases such an f f is unique and can be expressed as a fraction f = b / a f=b/a with a , b ∈ K a,b\in \mathcal {K} . In this paper, we study interpolants that are such fractions of K \mathcal {K} functions and are bounded in norm by 1 1 (assuming that ‖ T ‖ > 1 \|T\|>1 , in which case they always exist). We parameterize the collection of all such pairs ( a , b ) ∈ K × K (a,b)\in \mathcal {K}\times \mathcal {K} and show that each interpolant of this type can be determined as the unique minimum...
The problem of stabilizing a second order SISO LTI system of the form ẋ = Ax + Bu, y = Cx with fe... more The problem of stabilizing a second order SISO LTI system of the form ẋ = Ax + Bu, y = Cx with feedback of the form u(x) = v(x)Cx is considered, where v(x) is realvalued and has domain which is all of R 2 . It is shown that, when stabilization is possible, v(x) can be chosen to take on no more than two values throughout the entire state space (i.e., v(x) ∈ {v 1, v2} for all x and for some v1, v2), and an algorithm for finding a specific choice of v(x) is presented. It is also shown that the classical root locus of the corresponding transfer function C(sI -A) -1 B has a strong connection to this stabilization problem, and its utility is demonstrated through several design examples.
The motivation for this paper stems from the need to develop a uniform framework for addressing p... more The motivation for this paper stems from the need to develop a uniform framework for addressing problems in identiÿcation and robust control. System identiÿcation for inÿnite-dimensional Hilbert spaces has been addressed earlier by the authors. System identiÿcation set in an Hilbert space results in uncertain models where the description of non-parametric error is typically a ball belonging to the Hilbert space. The scope of this paper is to complement these results -develop robust control-synthesis and analysis results -for some special, yet, important cases. In this paper we derive a convex parameterization of robustly stabilizing controllers for LTI discrete-time systems deÿned on Hilbert spaces. The perturbations are of rank-one type having both real-parametric and non-parametric components. The parameterization allows for imposing other constraints to obtain meaningful performance from the controller. Analysis tools are also developed for robust stability under SISO block-diagonally structured perturbations. The robustness analysis problem reduces to a ÿnite-dimensional LMI veriÿcation which makes the procedure extremely e cient.
A new method, Fourier model reduction, for obtaining stable, accurate, low-order models of very l... more A new method, Fourier model reduction, for obtaining stable, accurate, low-order models of very large linear systems is presented. The technique draws on traditional control and dynamical system concepts and utilizes them in a way which is computationally very efficient. Discretetime Fourier coefficients of the large system are calculated and used to construct a reduced-order model that preserves stability properties of the original system. Many coefficients can be calculated, which results in a very accurate representation of the system dynamics, but only a single factorization of the large system is required. The resulting system can be further reduced using explicit formulae for balanced truncation. The method is applied to two computational fluid dynamic systems, which model unsteady motion of a two-dimensional subsonic airfoil and unsteady flow in a supersonic diffuser. In both cases, the new method is found to work extremely well. Results are compared to models developed using the proper orthogonal decomposition and Arnoldi method. In comparison with these widely used techniques, the new method is computationally more efficient, preserves the stability of the original system, uses both input and output information, and, for smooth transfer functions, is valid over a wide range of frequencies.
We develop a sufficient condition for the least-squares measurement (LSM), or the squareroot meas... more We develop a sufficient condition for the least-squares measurement (LSM), or the squareroot measurement, to minimize the probability of a detection error when distinguishing between a collection of mixed quantum states. Using this condition we derive the optimal measurement for state sets with a broad class of symmetries. We first consider geometrically uniform (GU) state sets with a possibly nonabelian generating group, and show that if the generator satisfies a certain constraint, then the LSM is optimal. In particular, for pure-state GU ensembles the LSM is shown to be optimal. For arbitrary GU state sets we show that the optimal measurement operators are GU with generator that can be computed very efficiently in polynomial time, within any desired accuracy. We then consider compound GU (CGU) state sets which consist of subsets that are GU. When the generators satisfy a certain constraint, the LSM is again optimal. For arbitrary CGU state sets the optimal measurement operators are shown to be CGU with generators that can be computed efficiently in polynomial time.
IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 2010
This paper presents a system identification technique for generating stable compact models of typ... more This paper presents a system identification technique for generating stable compact models of typical analog circuit blocks in radio frequency systems. The identification procedure is based on minimizing the model error over a given training data set subject to an incremental stability constraint, which is formulated as a semidefinite optimization problem. Numerical results are presented for several analog circuits, including a distributed power amplifier, as well as a MEM device. It is also shown that our dynamical models can accurately predict important circuit performance metrics, and may thus, be useful for design optimization of analog systems.
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Papers by A. Megretski