Performance Funnel Research Articles

Funnel Control for Linear DAEs

AbstractWe study funnel control for linear differential‐algebraic multi‐input multi‐output systems which are not necessarily regular. We show that the funnel controller (that is a static nonlinear output error feedback) achieves ‐ for a special class of right‐invertible systems with asymptotically stable zero dynamics ‐ tracking of a reference signal by the output signal within a pre‐specified performance funnel. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Funnel Flight Controller for a TDR 6-DoF Simulation Model

A Throttleable Ducted Rocket (TDR) is a ramjet-type engine using a solid propellant for which the fuel mass flow rate can be controlled. Since the propellant has an oxygen deficient formulation, fuel rich combustion products are produced within the gas generator which are exhausted into the ram combustor and are afterburned with air. Modeling is an extremely important tool for proper TDR design. For the TDR, this model has to represent the internal aerodynamics – including the two combustion processes within the gas generator and the ram combustor – accurately. The simulation of in-flight performance has been an important area throughout the history of missile development. The use of formal methods for analysis and predictability is essential, since it is impractical to cover all of the operational regime of an air-breathing propulsion system by ground tests and flight tests of the complete missile are extremely expensive. This paper investigates the trajectory prediction for a guided and thrust controlled ramjet missile. In this paper, a trajectory simulation model using a 6-DoF model was developed and applied for a typical TDR. A detailed description of the 6-DoF method is provided. An effort has been made to take into account the aerodynamic, gravitational and inertial coupling dynamics. This paper starts with a description of the general design of a TDR and the motor model. A brief introduction to the motor control algorithms is then provided. This is followed by a description of the mathematical 6-DoF trajectory simulation model including a flight controller – a so-called 4-dimensional performance funnel. For all models, a funnel controller method is used as an automatic optimization and control tool. In this paper, it is shown how the 6-DoF model of the TDR is used in a simulation for a military enemy trajectory track.

Impact of missing data on standardised mortality ratios for acute myocardial infarction: evidence from the Myocardial Ischaemia National Audit Project (MINAP) 2004–7

BackgroundStandardised mortality ratios (SMR) are often used to depict cardiovascular care. Data missingness, data quality, temporal variation and case-mix can, however, complicate the assessment of clinical performance.ObjectivesTo study Primary Care...

Funnel Control With Saturation: Linear MIMO Systems

Tracking-by the system output-of a reference signal (assumed bounded with essentially bounded derivative) is considered in the context of linear <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> -input, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> -output systems subject to input saturation. The system is assumed to have strict relative degree one with stable zero dynamics. Prespecified is a performance funnel, within which the tracking error is required to evolve: transient and asymptotic behavior of the tracking error is influenced through choice of parameter values which define the funnel. The control structure is a saturating error feedback with a gain function designed to evolve so as to preclude contact with the funnel boundary. A feasibility condition (formulated in terms of the system data, the saturation bounds, the funnel data, bounds on the reference signal, and the initial state) is presented under which the tracking objective is achieved, whilst maintaining boundedness of all signals.

Tracking with Prescribed Transient Performance for Hysteretic Systems

Tracking of reference signals (assumed bounded with essentially bounded derivative) is considered for a class of single-input, single-output, nonlinear systems, described by a functional differential equation with a hysteresis nonlinearity in the input channel. The first control objective is tracking, by the output, with prescribed accuracy: determine a feedback strategy which ensures that, for every reference signal and every system of the underlying class, the tracking error ultimately satisfies the prescribed accuracy requirements. The second objective is guaranteed output transient performance: the graph of the tracking error should be contained in a prescribed set (performance funnel). Under a weak sector boundedness assumption on the hysteresis operator, both objectives are achieved by a memoryless feedback which is universal for the underlying class of systems.

Controlled functional differential equations: approximate and exact asymptotic tracking with prescribed transient performance

A tracking problem is considered in the context of a class S of multi-input, multi-output, nonlinear systems modelled by controlled functional differential equations. The class contains, as a prototype, all finite-dimensional, linear, m-input, m-output, minimum-phase systems with sign-definite high-frequency gain. The first control objective is tracking of reference signals r by the output y of any system in S :g ivenλ ≥ 0, construct a feedback strategy which ensures that, for every r (assumed bounded with essentially bounded derivative) and every system of class S, the tracking error e = y − r is such that, in the case λ> 0, lim supt→∞ � e(t)� <λ or, in the case λ = 0, limt→∞ � e(t)� =0 . The second objective is guaranteed output transient performance: the error is required to evolve within a prescribed performance funnel Fϕ (determined by a function ϕ). For suitably chosen functions α, ν and θ, both objectives are achieved via a control structure of the form u(t )= −ν(k(t))θ(e(t)) with k(t )= α(ϕ(t)� e(t)� ), whilst maintaining boundedness of the control and gain functions u and k .I n the case λ = 0, the feedback strategy may be discontinuous: to accommodate this feature, a unifying framework of differential inclusions is adopted in the analysis of the general case λ ≥ 0.

Tracking with Prescribed Transient Behavior for Nonlinear Systems of Known Relative Degree

Tracking of a reference signal (assumed bounded with essentially bounded derivative) is considered in the context of a class $\Sigma_{\rho}$ of multi‐input, multi‐output dynamical systems, modelled by functional differential equations, affine in the control and satisfying the following structural assumptions: (i) arbitrary—but known—relative degree $\rho \ge 1; (ii) the “high‐frequency gain” is sign definite—but possibly of unknown sign. The class encompasses a wide variety of nonlinear and infinite‐dimensional systems and contains (as a prototype subclass) all finite‐dimensional, linear, m‐input, m‐output, minimum‐phase systems of known strict relative degree. The first control objective is tracking, by the output y, with prescribed accuracy: given $\lambda >0$ (arbitrarily small), determine a feedback strategy which ensures that, for every reference signal r and every system of class $\Sigma_{\rho}$, the tracking error $e=y-r$ is ultimately bounded by λ (that is, $\|e(t)\| < \lambda$ for all t sufficiently large). The second objective is guaranteed output transient performance: the tracking error is required to evolve within a prescribed performance funnel $\mathcal{F}_\varphi$ (determined by a function φ). Both objectives are achieved using a filter in conjunction with a feedback function of the tracking error, the filter states, and the funnel parameter φ.

Asymptotic tracking with prescribed transient behaviour for linear systems

The problem of asymptotic tracking of reference signals is considered in the context of m-input, m-output linear systems (A,B,C) with the following structural properties: (i) CB is sign definite (but not necessarily symmetric), (ii) the zero dynamics are exponentially stable. The class of reference signals is the set of all possible solutions of a fixed, stable, linear, homogeneous differential equation (with associated characteristic polynomial α). The first control objective is asymptotic tracking, by the system output y = Cx, of any reference signal . The second objective is guaranteed error e = y − r transient performance: e should evolve within a prescribed performance funnel (determined by a function ϕ). Both objectives are achieved simultaneously by an internal model in series with a proportional time-varying error feedback t↦ u(t)=−ν (k(t)) e(t), where ν is a smooth function with the properties and , and k(t) is generated via a nonlinear function of the product ‖ e(t)‖ ϕ (t) The feedback structure essentially exploits an intrinsic high-gain property of the system by ensuring that, if (t,e(t)) approaches the funnel boundary, then the gain attains values sufficiently large to preclude boundary contact.

Tracking control with prescribed transient behaviour for systems of known relative degree

Tracking of a reference signal (assumed bounded with essentially bounded derivative) is considered for multi-input, multi-output linear systems satisfying the following structural assumptions: (i) arbitrary—but known—relative degree, (ii) the “high-frequency gain” matrix is sign definite—but of unknown sign, (iii) exponentially stable zero dynamics. The first control objective is tracking, by the output y , with prescribed accuracy: given λ > 0 (arbitrarily small), determine a feedback strategy which ensures that, for every reference signal r, the tracking error e = y - r is ultimately bounded by λ (that is, ∥ e ( t ) ∥ < λ for all t sufficiently large). The second objective is guaranteed output transient performance: the evolution of the tracking error should be contained in a prescribed performance funnel F ϕ (determined by a function ϕ ). Both objectives are achieved by a filter in conjunction with a feedback function of the filter states, the tracking error and a gain parameter. The latter is generated via a feedback function of the tracking error and the funnel parameter ϕ . Moreover, the feedback system is robust to nonlinear perturbations bounded by some continuous function of the output. The feedback structure essentially exploits an intrinsic high-gain property of the system/filter interconnection by ensuring that, if ( t , e ( t ) ) approaches the funnel boundary, then the gain attains values sufficiently large to preclude boundary contact.

Tracking control: Performance funnels and prescribed transient behaviour

Tracking of a reference signal (assumed bounded with essentially bounded derivative) is considered in the context of a class of nonlinear systems, with output y, described by functional differential equations (a generalization of the class of linear minimum-phase systems of relative degree one with positive high-frequency gain). The primary control objective is tracking with prescribed accuracy: given λ > 0 (arbitrarily small), determine a feedback strategy which ensures that for every admissible system and reference signal, the tracking error e = y - r is ultimately smaller than λ (that is, ∥ e ( t ) ∥ < λ for all t sufficiently large). The second objective is guaranteed transient performance: the evolution of the tracking error should be contained in a prescribed performance funnel F . Adopting the simple non-adaptive feedback control structure u ( t ) = - k ( t ) e ( t ) , it is shown that the above objectives can be attained if the gain is generated by the nonlinear, memoryless feedback k ( t ) = K F ( t , e ( t ) ) , where K F is any continuous function exhibiting two specific properties, the first of which ensures that if ( t , e ( t ) ) approaches the funnel boundary, then the gain attains values sufficiently large to preclude boundary contact, and the second of which obviates the need for large gain values away from the funnel boundary.

Input constrained funnel control with applications to chemical reactor models

Error feedback control (in the presence of input constraints) is considered for a class of exothermic chemical reactor models. The primary control objective is regulation of a setpoint temperature T ∗ with prescribed accuracy: given λ>0 (arbitrarily small), ensure that, for every admissible system and reference setpoint, the regulation error e=T−T ∗ is ultimately smaller than λ (that is, || e( t)||< λ for all t sufficiently large). The second objective is guaranteed transient performance: the evolution of the regulation error should be contained in a prescribed performance funnel F around the setpoint temperature T ∗ . A simple error feedback control with input constraints of the form u(t)= sat [ u ̄ , u ̄ ] (−k(t)[T(t)−T ∗]+u ∗) , u ∗ an offset, is introduced which achieves the objective in the presence of disturbances corrupting the measurement. The gain k( t) is a function of the error e(t)=T(t)−T ∗ and its distance to the funnel boundary. The input constraints u ̄ , u ̄ have to satisfy certain feasibility assumptions in terms of the model data and the operating point T ∗ .