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The present note is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021) on generic controllability and stabilizability of linear differential-algebraic equations. We resolve the drawback that genericity is... more
The present note is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021) on generic controllability and stabilizability of linear differential-algebraic equations. We resolve the drawback that genericity is considered in the unrestricted set of system matrices $$(E,A,B)\in \mathbb {R}^{\ell \times }\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}$$ ( E , A , B ) ∈ R ℓ × × R ℓ × n × R ℓ × m , while for relative genericity we allow the restricted set $$\Sigma _{\ell ,n,m}^{\le r} := \{(E,A,B)\in \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m} \,\big \vert \,\textrm{rk}\,_{\mathbb {R}} E \le r\}$$ Σ ℓ , n , m ≤ r : = { ( E , A , B ) ∈ R ℓ × n × R ℓ × n × R ℓ × m | rk R E ≤ r } , where $$ r\in {\mathbb {N}}$$ r ∈ N . Our main results are characterizations of generic controllability and generic stabilizability in $$\Sigma _{\ell ,n,m}^{\le r}$$ Σ ℓ , n , m ≤ r in terms of the numbers $$\ell ...
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We investigate genericity of various controllability and stabilizability concepts of linear, time-invariant differential-algebraic systems. Based on well-known algebraic characterizations of these concepts (see the survey article by... more
We investigate genericity of various controllability and stabilizability concepts of linear, time-invariant differential-algebraic systems. Based on well-known algebraic characterizations of these concepts (see the survey article by Berger and Reis (in: Ilchmann A, Reis T (eds) Surveys in differential-algebraic equations I, Differential-Algebraic Equations Forum, Springer, Berlin, pp 1–61. 10.1007/978-3-642-34928-7_1)), we use tools from algebraic geometry to characterize genericity of controllability and stabilizability in terms of matrix formats.
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For linear time-varying systems with bounded system matrices we discuss the problem of stabilizability by linear state feedback. It is shown that an optimal control approach yields a criterion in terms of the cost for stabilizability. The... more
For linear time-varying systems with bounded system matrices we discuss the problem of stabilizability by linear state feedback. It is shown that an optimal control approach yields a criterion in terms of the cost for stabilizability. The constants appearing in the criterion of optimality allow for the distinction of exponential and uniform exponential stabilizability. We show that the system is completely controllable if, and only if, the Lyapunov exponent is arbitrarily assignable by a suitable feedback. For uniform exponential stabilizability and the assignability of the Bohl exponent this property is known. Also, dynamic feedback does not provide more freedom to address the stabilization problem.
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In this paper we present three applications of adaptive X- tracking that show the suitability of this control method for industrial process control problems. The X-tracker is a simple design/low complexity adaptive controller that is... more
In this paper we present three applications of adaptive X- tracking that show the suitability of this control method for industrial process control problems. The X-tracker is a simple design/low complexity adaptive controller that is universal in the sense, that no process model or plant tests are required for the controller design. Furthermore, the controller exhibits remarkable robustness properties. We briefly review the theory of adaptive X-tracking, and discuss advantages and shortcomings of this method. formance or even instability may occur. In any case, there is no assurance for the stability of the closed loop other than experience based on similar control problems from the past. In this paper we discuss the application of the recently developed adaptive A-tracking controller ("adaptive X- tracker") to industrial process control problems. This con- troller combines simple design and implementation with novel control theoretic methods that allow to guarantee stabil...
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Research Interests: Engineering, Mechanical Engineering, Mathematics, Applied Mathematics, Environmental Science, and 10 moreComputer Science, Adaptive Control, Tracking, Article, True, Yeast fermentation, Continuous stirred tank reactor, Electrical And Electronic Engineering, Ordinary Differential Equation, and Input Saturation
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Research Interests: Mathematics, Adaptive Control, Tracking, Adaptive, Article, and 5 moreRobustness, True, High Gain, Observers, and Degree in music
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We discuss the concept of `minimum phase' for scalar semi Hurwitz transfer functions. The latter are rational functions where the denominator polynomial has its roots in the closed left half complex plane. In the present note, minimum... more
We discuss the concept of `minimum phase' for scalar semi Hurwitz transfer functions. The latter are rational functions where the denominator polynomial has its roots in the closed left half complex plane. In the present note, minimum phase is defined in terms of the derivative of the argument function of the transfer function. The main tool to characterize minimum phase is the Hurwitz reflection. The factorization of a weakly stable transfer function into an all-pass and a minimum phase system leads to the result that any semi Hurwitz transfer function is minimum phase if, and only if, its numerator polynomial is semi Hurwitz. To characterize the zero dynamics, we use the Byrnes-Isidori form in the time domain and the internal loop form in the frequency domain. The uniqueness of both forms is shown. This is used to show in particular that asymptotic stable zero dynamics of a minimal realization of a transfer function yields minimum phase, but not vice versa.
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We study positive linear Volterra integro-difierential systems with inflnitely many delays. Positivity is characterized in terms of the system en- tries. A generalized version of the Perron-Frobenius Theorem is shown; this may be... more
We study positive linear Volterra integro-difierential systems with inflnitely many delays. Positivity is characterized in terms of the system en- tries. A generalized version of the Perron-Frobenius Theorem is shown; this may be interesting in its own right but is exploited here for stability results: explicit spectral criteria for L1-stability and exponential asymptotic stability. Also the concept of stability radii, determining the maximal robustness with respect to additive perturbations to L1-stable system, is introduced and it is shown that the complex, real and positive stability radii coincide and can be computed by an explicit formula.
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In this paper we show that well-known high-gain universal adaptive P-controllers can be implemented digitally, via adaptive sampling, provided that the length of the sampling interval increases sufficiently fast, as the proportional gain... more
In this paper we show that well-known high-gain universal adaptive P-controllers can be implemented digitally, via adaptive sampling, provided that the length of the sampling interval increases sufficiently fast, as the proportional gain increases. Both stabilization and λ-tracking of arbitrary bounded and essentially smooth reference signals are considered.
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We consider adaptive regulation of the temperature for exothermic chemical reactors. These reactors may exhibit multiple steady states, and the relevant operating point could be an unstable open-loop equilibrium. The adaptive controller... more
We consider adaptive regulation of the temperature for exothermic chemical reactors. These reactors may exhibit multiple steady states, and the relevant operating point could be an unstable open-loop equilibrium. The adaptive controller combines the so called À-tracking approach with a feedback which obeys a saturation; it is simple in its design, does not invoke an observer, and can cope with measurement noise. Tracking is achieved up to any pre-specified accuracy, the choice of which is left to the designer. The class of exothermic chemical reactors to which the controller is applicable is considerably larger than the classes to which robust (non-adaptive) global stabilisers, previously studied in the literature, can be applied. The adaptive control strategy does not require any knowledge of the systems parameters and does not invoke an internal model. It is only assumed that the reference temperature to be tracked is feasible. The problem of temperature tracking is solved both l...
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Tracking of reference signals is addressed in the context of a class of nonlinear controlled systems modelled by r-th-order functional differential equations, encompassing inter alia systems with unknown “control direction” and dead-zone... more
Tracking of reference signals is addressed in the context of a class of nonlinear controlled systems modelled by r-th-order functional differential equations, encompassing inter alia systems with unknown “control direction” and dead-zone input effects. A control structure is developed which ensures that, for every member of the underlying system class and every admissible reference signal, the tracking error evolves in a prescribed funnel chosen to reflect transient and asymptotic accuracy objectives. Two fundamental properties underpin the system class: bounded-input bounded-output stable internal dynamics, and a high-gain property (an antecedent of which is the concept of sign-definite high-frequency gain in the context of linear systems).
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ABSTRACT We consider adaptive regulation of the temperature for exothermic chemical reactors. These reactors may exhibit multiple steady states, and the relevant operating point could be an unstable open-loop equilibrium. The adaptive... more
ABSTRACT We consider adaptive regulation of the temperature for exothermic chemical reactors. These reactors may exhibit multiple steady states, and the relevant operating point could be an unstable open-loop equilibrium. The adaptive controller combines the so called -tracking approach with a feedback which obeys a saturation; it is simple in its design, does not invoke an observer, and can cope with measurement noise. Tracking is achieved up to any pre-specified accuracy, the choice of which is left to the designer. The class of exothermic chemical reactors to which the controller is applicable is considerably larger than the classes to which robust (non-adaptive) global stabilisers, previously studied in the literature, can be applied. The adaptive control strategy does not require any knowledge of the systems parameters and does not invoke an internal model. It is only assumed that the reference temperature to be tracked is feasible. The problem of temperature tracking is solved both locall...
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We consider adaptive regulation of the temperature for exothermic chemical reactors. These reactors may exhibit multiple steady states, and the relevant operating point could be an unstable open-loop equilibrium. The adaptive controller... more
We consider adaptive regulation of the temperature for exothermic chemical reactors. These reactors may exhibit multiple steady states, and the relevant operating point could be an unstable open-loop equilibrium. The adaptive controller combines the so called -tracking approach with a feedback which obeys a saturation; it is simple in its design, does not invoke an observer, and can cope with measurement noise. Tracking is achieved up to any pre-specified accuracy, the choice of which is left to the designer. The class of exothermic chemical reactors to which the controller is applicable is considerably larger than the classes to which robust (non-adaptive) global stabilisers, previously studied in the literature, can be applied. The adaptive control strategy does not require any knowledge of the systems parameters and does not invoke an internal model. It is only assumed that the reference temperature to be tracked is feasible. The problem of temperature tracking is solved both loc...
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Preprint / Technische Universität Ilmenau, Institut für Mathematik ; 10-07 Abstact: We introduce a solution theoryfor time-varying linear differential-algebraic equations(DAEs) E(t)˙x = A(t)x which can be transformed into standard... more
Preprint / Technische Universität Ilmenau, Institut für Mathematik ; 10-07 Abstact: We introduce a solution theoryfor time-varying linear differential-algebraic equations(DAEs) E(t)˙x = A(t)x which can be transformed into standard canonical form (SCF), i.e. the DAE is decoupled into an ODE ˙z1 = J(t)z1 and a pure DAE N(t)˙z1 = z1 , where N is pointwise strictly lower triangular. This class is a time-varying generalization of time-invariant DAEs where the corresponding matrix pencil is regular. It will be shown in which sense the SCF is a canonical form, that it allows for a transition matrix similar to the one for ODEs, and how this can be exploited to derive a variation of constants formula. Furthermore, we show in which sense the class of systems transferable into SCF is equivalent to DAEs which are analytically solvable, and relate SCF to the derivative array approach, differentiation index and strangeness index. Finally, an algorithm is presented which determines the transformatio...
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ABSTRACT We consider linear differential-algebraic m-input m-output systems with positive strict relative degree or proper inverse transfer function; in the single-input single-output case these two disjoint classes make the whole of all... more
ABSTRACT We consider linear differential-algebraic m-input m-output systems with positive strict relative degree or proper inverse transfer function; in the single-input single-output case these two disjoint classes make the whole of all linear DAEs without feedthrough term. Structural properties - such as normal forms (i.e. the counterpart to the Byrnes-Isidori form for ODE systems), zero dynamics, and high-gain stabilizability - are analyzed for two purposes: first, to gain insight into the system classes and secondly, to solve the output regulation problem by funnel control. The funnel controller achieves tracking of a class of reference signals within a pre-specified funnel; this means in particular, the transient behaviour of the output error can be specified and the funnel controller does neither incorporate any internal model for the reference signals nor any identification mechanism, it is simple in its design. The results are illuminated by position and velocity control of a mechanical system encompassing springs, masses, and dampers.
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We introduce a solution theory for time-varying linear differential-algebraic equations (DAEs) E ( t ) x ˙ = A ( t ) x E(t)\dot x=A(t)x which can be transformed into standard canonical form (SCF); i.e., the DAE is decoupled into an ODE z... more
We introduce a solution theory for time-varying linear differential-algebraic equations (DAEs) E ( t ) x ˙ = A ( t ) x E(t)\dot x=A(t)x which can be transformed into standard canonical form (SCF); i.e., the DAE is decoupled into an ODE z ˙ 1 = J ( t ) z 1 \dot z_1 = J(t)z_1 and a pure DAE N ( t ) z ˙ 2 = z 2 N(t) \dot z_2 = z_2 , where N N is pointwise strictly lower triangular. This class is a time-varying generalization of time-invariant DAEs where the corresponding matrix pencil is regular. It will be shown in which sense the SCF is a canonical form, that it allows for a transition matrix similar to the one for ODEs, and how this can be exploited to derive a variation of constants formula. Furthermore, we show in which sense the class of systems transferable into SCF is equivalent to DAEs which are analytically solvable, and relate SCF to the derivative array approach, differentiation index and strangeness index. Finally, an algorithm is presented which determines the transforma...
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Research Interests: Mathematics, Applied Mathematics, Operator Theory, Potential Theory, Pure Mathematics, and 13 moreMathematical Analysis, Article, Convolution, Positivity, Linear Stability, True, DDC, Singular Volterra Integral Equation, Volterra integral equations, Asymptotic Stability, Linear Time, Integro Differential Equation, and Exponential Stability
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For linear time-varying systems with bounded system matrices we discuss the problem of stabilizability by linear state feedback. For example, it is shown that complete controllability implies the existence of a feedback so that the... more
For linear time-varying systems with bounded system matrices we discuss the problem of stabilizability by linear state feedback. For example, it is shown that complete controllability implies the existence of a feedback so that the closed-loop system is asymptotically stable. We also show that the system is completely controllable if, and only if, the Lyapunov exponent is arbitrarily assignable by a suitable feedback. For uniform exponential stabilizability and the assignability of the Bohl exponent this property is known. Also, dynamic feedback does not provide more freedom to address the stabilization problem. The unifying tools for our results are two finite (L2L2) cost conditions. The distinction of exponential and uniform exponential stabilizability is then a question of whether the finite cost condition is uniform in the initial time or not.