VIBRATION DAMPER

Technical Field of the Invention

The invention relates to an active absorber for an AC electricity generating engine particularly incorporating a linear alternator, an engine system comprising such an active absorber (in which the engine system may be a combined heat and power unit) and a method for active damping of an AC electricity generating engine.

Background to the Invention

Engine-based systems are always subjected to vibrations and, damping of such vibrations is a particular problem for electricity generating engines. For example, Free Piston Stirling Engine Generators are synchronous machines with a linear alternator (FPSE/LA). The vibrations are typically due to the reciprocating motion of permanent magnets attached to the power piston. This motion results in a reaction force according to Lorentz law which acts on the alternator coil along with dynamic forces from the moving parts. This alternator is attached to the engine's case, so that this reaction force is the main cause of vibrations.

For a reliable operation of engines such as the FPSE/LA, a tuned mass-spring absorber may be fitted to minimize engine casing vibrations. A passive absorber, such as a tuned-mass damper (TMD) tuned to damp vibration at 50Hz or 60Hz, has conventionally been used. These are simple and reliable, but have performance limitations, for instance in terms of their narrow bandwidth. An improved technique for vibration damping that overcomes these limitations is therefore desirable. Summary of the Invention

Against this background, there is provided an active absorber for a system comprising an AC electricity generating engine (particularly with a linear alternator), comprising: a sensor, arranged to determine (or measure) at least one parameter of the system; and an actuator, configured to provide an adjustable force. The force is set on the basis of the at least one determined parameter of the system. The AC electricity generating engine may be a Free Piston Stirling Engine (FPSE).

The actuator may therefore be part of an active damper or absorber. The actuator may comprise a hydraulic piston, a piezoelectric device, an electric motor or similar technology. The active damper or absorber may be tuneable to a system frequency, which may vary. Although a passive absorber can deal with a range of vibration frequencies, this range is typically narrow. An active absorber can cope with a wider frequency range. Existing types of generator are designed and tuned to operate within a frequency range of frequency 50Hz or 60Hz ± 0.5Hz. A wider frequency operating range (47Hz-52Hz) may be desirable though. High vibration amplitudes encountered over this frequency range may lead to unacceptable noise and reliability issues. An active absorber may address this problem.

Preferably, the actuator is configured to form part of a tuned mass damper having at least one gain. In particular, an active tuned-mass damper may be created by connecting a passive tuned-mass damper to an actuator. The actuator may then be operated through a suitable control strategy based on the measurements and responses of the system to be damped. The tuned mass damper gain may be defined as the factor relating the actuator force to the system state. The system state is typically defined by the position and/or motion of at least one part of the tuned mass damper. These may define one or more dynamic parameters of the system, as will be discussed below. There may be multiple tuned mass damper gains, each tuned mass damper gain relating to a respective variable of the system state. In particular, the active absorber may further comprise a sensor arrangement configured to determine (or measure) the system state.

In some embodiments, the sensor is arranged to determine an electrical frequency associated with the electricity generating engine. Then, the at least one gain may be adjusted on the basis of the electrical frequency. This may be understood as a feed forward approach. The electrical frequency may comprise one of: the AC electricity generated by the engine; and AC electricity coupled to the AC electricity generated by the engine.

Advantageously, the at least one gain is adjusted in order to set a resonant frequency of the tuned mass damper to match the electricity frequency. In some embodiments, the gain is adjusted on the basis of an incremental change in the electricity frequency, u) _inc . The tuned mass damper may have a resonant frequency, ooorig. The gain may then be further adjusted to set a new resonant frequency of the tuned mass damper, u) _ne w The new resonant frequency of the tuned mass damper may be set to be the sum of the resonant frequency, ω _0Γ , ₉ , and the incremental change in the electricity frequency, o , i.e. ω _0Γ ί ₉ + u) _inc . The active absorber may further comprise a sensor arrangement configured to determine (or measure) ω _0Γ ί ₉ or ω,ηο or both.

In the preferred embodiment, the sensor is arranged to determine at least one dynamic parameter of the system. In this case, the at least one associated gain may be adjusted on the basis of the at least one dynamic parameter. A feedback controller based on a comparison of each of the at least one dynamic parameter with a reference value, such as a PID controller, may be provided in combination with a feed forward controller. In some embodiments, the force provided by the actuator may be set on the basis of a Linear Quadratic, LQ, control. A feed forward approach is not necessary for this type of control and the same may be true of other types of control.

A model for the tuned mass damper is well known. Typically, the active absorber further comprises an absorber dynamic mass. The force (provided by the actuator) may then be set on the basis of the at least one dynamic parameter. Beneficially, the at least one dynamic parameter comprises one or both of: a position, x ₂ ; and an acceleration, x ₂ , of the absorber dynamic mass. In particular, the force may be set on the basis of at least one relative motion parameter. The at least one relative motion parameter may comprise one or both of: the position of the absorber dynamic mass, x ₂ , relative to a position of the engine, xi (which may be expressed as x ₂ - xi); and the acceleration of the absorber dynamic mass, x ₂ , relative to an acceleration of the engine, xi (which may be expressed as x ₂ - xi). Then, the force may be set on the basis of the product of the at least one relative motion parameter and the tuned mass damper gain. The active absorber may further comprise a sensor arrangement configured to determine (or measure) the at least one motion parameter.

Preferably, the active absorber further comprises a coupling arrangement, arranged to couple the absorber dynamic mass to the engine. The coupling

arrangement may have a spring constant (or stiffness), k ₂ . Then, the force may further be set on the basis of this spring constant. The model for the tuned mass damper typically defines the vibrating object coupled to a reference surface by a first elastic element (normally modelled as a spring) and a mass coupled to the vibrating object by a second elastic element (normally also modelled as a spring). Thus, the force may further be set on the basis of the spring constant for the second elastic element, but not necessarily any parameter of the first elastic element. Additionally or alternatively, the force may be further set on the basis of the mass of the absorber dynamic mass, m ₂ . However, the force need not be set on the basis of the mass of the engine, m .

When the force is set on the basis of the position of the absorber dynamic mass, x ₂ , relative to a position of the engine, Xi , the tuned mass damper gain may comprise a first tuned mass damper gain, defined as γ. Then, the tuned mass damper gain may comprise γ = m^o + ω _ΟΓ ί ₉ ) ² - k ₂ . Additionally or alternatively, the force may be set on the basis of the acceleration of the absorber dynamic mass, x ₂ , and on the basis of the acceleration of the engine, xi . Then, the tuned mass damper gain may comprise a second tuned mass damper gain, defined as a. In this case, the tuned mass damper gain may comprise a = k ₂ / (u) _inc +

In the preferred embodiment, the active absorber further comprises a controller, configured to calculate at least one gain associated with the tuned mass damper.

Optionally, the controller is then configured to determine a force for the actuator on the basis of the calculated tuned mass damper gain. Then, the controller may then be configured to control the actuator to provide the calculated force. In some embodiments, the controller comprises a feed forward controller portion. In embodiments, the controller comprises: a first, feed forward controller portion; and a second, feedback controller portion. The second controller portion may comprise a PID controller. In some embodiments, the controller is a Linear Quadratic, LQ, controller. Then, the controller may comprise a Linear Quadratic, LQ, control part. The at least one associated gain may be adjusted on the basis of the at least one dynamic parameter. Where the sensor is arranged to determine the position and acceleration of the absorber mass and the engine, the LQ control part may be termed a LQ controller. Optionally, the controller further comprises a state space estimator. Where the sensor is arranged to determine relative parameters, the state space estimator may be used to determine inputs to the LQ control part. This may be termed a Linear Quadratic Gaussian LOG controller.

Preferably, the active absorber is configured for use with a Stirling engine, particularly a β-type Stirling engine. More preferably, the active absorber is configured for use with a Free Piston Stirling Engine Generator with a linear alternator (FPSE/LA). The active absorber is beneficially configured for use with a synchronous engine.

In the preferred embodiment, the active absorber is configured for use with an engine of a combined heat and power unit, for example a domestic combined heat and power (DCHP) unit. A combined heat and power unit may also be provided, comprising such an active absorber.

In another aspect, there is provided an engine system, comprising: an engine, configured for generating AC electricity; and the active absorber as described herein, coupled to the engine. The engine may be configured for connection into an external AC electricity supply. The engine may be a synchronous engine. It may then be configured to set the frequency of the AC electricity generated by the engine to match the frequency of the electricity from the external AC electricity supply. As noted above, the engine system may be a combined heat and power unit, particularly a domestic combined heat and power (DCHP) unit.

In a yet further aspect, there is provided a method for active damping of an AC electricity generating engine as part of a system (particularly with a linear alternator), comprising: determining at least one parameter of the system; and setting a force to be provided by an actuator, the force being determined on the basis of the at least one parameter. Preferably, the actuator forms part of a tuned mass damper, advantageously coupled to the engine. Beneficially, the method further comprises: controlling the actuator to apply the set force. Additionally or alternatively, the method may comprise providing the set force using the actuator. Method features corresponding with any of the structural or apparatus features described herein may further be optionally provided.

In another aspect, there is provided a computer program, configured to carry out any method as described herein, when operated by a processor. The invention may also be embodied in logic, such as digital or programmable logic, configured to carry out any method as described herein.

There may also be provided any combination of specific structural, apparatus or method features detailed herein, whether or not such a combination is explicitly described. Brief Description of the Drawings

The invention may be put into practice in various ways, one of which will now be described by way of example only and with reference to the accompanying drawings in which :

Figure 1 depicts a schematic diagram of a known passive tuned mass damper; Figure 2 illustrates a schematic diagram of an existing engine system, using a passive tuned mass damper in accordance with that shown in Figure 1 ;

Figures 3A to 3C show frequency responses for a simulation of an engine system in accordance with Figure 2;

Figure 4 is a plot of acceleration against frequency from experimental results for a number of points on an engine configured in accordance with Figure 2;

Figure 5A shows a schematic diagram of a known active tuned mass damper, using similar features to those of Figure 1 ;

Figure 5B shows a schematic diagram of an engine system in accordance with the invention, using an active tuned mass damper;

Figure 5C depicts a simplified model of the system of Figure 5B; Figure 6A depicts a model of an engine system and first controller in accordance with Figure 5B;

Figure 6B depicts an alternative model of an engine system and a first controller in accordance with Figure 5C;

Figure 7 A is a plot of position feedback gain against frequency increment for a simulation based on the model of Figure 6;

Figure 7B is a plot of acceleration feedback gain against frequency increment for a simulation based on the model of Figure 6;

Figure 8 illustrates a model of an engine system and a second controller in accordance with Figure 5C;

Figure 9A shows a block diagram representation of a state space model based on the model of Figure 6A;

Figure 9B shows the block diagram representation of Figure 9A with the addition of a third controller;

Figure 10 depicts a model of an engine system and the third controller in accordance with the invention, based on the model of Figure 5B;

Figure 1 1 shows a schematic diagram showing a superposition relationship for the third controller;

Figure 12 depicts the block diagram representation of Figure 9A with the addition of a fourth controller; and

Figure 13 depicts the block diagram representation of Figure 9A with the addition of a fifth controller.

Detailed Description of a Preferred Embodiment

Before describing an active absorber in accordance with the invention, it is useful to present a model, simulation results and experimental results of an existing passive absorber. This may assist in illustrating the benefits and configuration of the invention.

Passive Absorber Model

Referring first to Figure 1 , there is depicted a schematic diagram of a known passive Tuned Mass Damper (TMD). The working principle of the passive tuned-mass damper is based on coupling an auxiliary mass to a structure whose vibration to be minimised. For the tuned-mass damper to function properly there is desirably synchronization between the natural frequency of the added mass and the excitation frequency of the original structure. The auxiliary mass desirably exerts an inertial force 90 degrees out of phase with respect to displacement of the main structure.

In the model shown in Figure 1 , mass m ₂ is an auxiliary mass attached to machine of mass ΓΤΗ through a spring of stiffness k ₂ , and a damper element of damping coefficient c ₂ . The combined system m^+m ₂ is isolated to a fixed end via two springs of stiffness each.

Simulation results of such a model show that, while there exists no damping in the absorber mass, the amplitude of the main mass shoots towards infinity at the natural frequencies of the combined system. The amplitude of the main mass is attenuated to zero at the natural frequency of the tuned-mass damper though. In other words, this can be considered as an ideal case scenario where the tuned-mass damper can completely minimise the vibration of the primary mass at resonance. However, the risk of this should be considered, as a little drift in the excitation frequency may cause devastating results, due to the poor bandwidth of the absorber. As the damping coefficient is increased, the amplitude of the primary mass displacement can no longer be attenuated to zero magnitude at the resonance frequency of the tuned-mass damper. As the mass ratio is increased, the natural frequencies of the combined system move further apart. This is useful as damping could then be reduced to improve the minimisation, where a little drift in the excitation frequency may not cause a sudden resonance on the primary mass.

An existing TMD has been applied to a system based on a Stirling engine.

Stirling engines are power machines that operate over a closed, regenerative

thermodynamic cycle, with cyclic compression and expansion of the working fluid at different temperature levels. A variety of heat sources can be utilized including solar energy, waste heat, and fossil fuels. The working fluid in Stirling engines might be air, nitrogen, helium, or hydrogen.

Referring now to Figure 2, there is illustrated a schematic diagram of an existing engine system, using a passive tuned mass damper in accordance with that shown in Figure 1 . This is a simplified, dynamic model. A Stirling engine is a complex machine and its behavioural description may require a multidisciplinary approach. An accurate

Free Piston Stirling Engine with Linear Alternator (FPSE/LA) dynamic model involves a complex multiphysics problem that couples thermodynamics, electromagnetism and mechanics. However, this type of model can be cumbersome to handle. Therefore, the use of simpler models is desirable from a dynamic point of view. The first subsystem a in Figure 2 represents the casing of the Stirling engine which forms the primary mass of the system. This primary mass contains a displacer and a piston that can be modelled as a spring-mass-damper system. The displacement of the piston causes the excitation which generates the vibration of the primary mass. By neglecting the internal dynamics of the piston and the displacer and by assuming that the piston is causing a sinusoidal excitation in the casing, a reasonable assumption of the Stirling engine system can be made by modelling it as a two degree of freedom (2DoF) system. The model is formed of a primary mass m , representing the engine case connected by a spring and damper to ground on one end and to a passive absorber on the other end similar to the model of Figure 1 with adding damping c ₂ to the primary mass. The equations of motion that describe the 2DoF system can be represented in a matrix form as the MIMO (multi-input multi-output) system shown below.

( [m]x(t) + [c]x(t) + [k]x(t) = F(t)

The variables x(t) and F (t) are called displacement and force vectors and are given by the following.

A more compact representation of a MIMO system can be defined in the form of a state-space model as follows.

The states x _x x _2t x ₃ and x ₄ represent the primary mass and absorber mass displacements and velocities respectively. The natural frequencies of the combined system in its first mode and the absorber mass can be written as follows respectively.

It follows that the general equation describing the displacements of the masses at steady state can be expressed in this form

- ηι ₂ ω ² ) + c ₂ a)j]F ₀

Xi =

where

[A = [(-m _j w ² + /c ₁ )(-m ₂ to ² + k ₂ ) - m ₂ k ₂ a) ² ] ² + [(/ - (m-^ + τη ₂ )ω ² ε ₂ ω)] j ω: excitation frequency

j ² = -i

For the particular engine used in this work, the system is subjected to a sinusoidal excitation force F(t) of magnitude 1000N and frequency 50Hz. The passive tuned-mass absorber is tuned to minimise the vibration of the primary mass at 50Hz. The first mode of vibration of the combined system corresponds to the free vibration of the system and has a natural frequency at about 2Hz. The simulation of the model is done in a Matlab (RTM) environment and the following table represents the simulation parameters.

Simulation parameters

m ₁ = 41.38 kg

m ₂ = 8.862 kg

c ₁ = 300 Ns/m

c ₂ = 2.7 Ns/m

k _t = 8369.4 N/m

k ₂ = 874640N/m

^combined ⁼ 4π rad/ S

ω _α = 100π rad/s

F(t) = 1000 sin ΙΟΟτζΐ N

Referring to Figure 3A, there is shown a frequency response for this simulation, showing the primary mass placement magnitude and phase against frequency (Bode plot). This Bode plot shows that the primary mass displacement has a peak at 2Hz due to resonance of the combined system at this frequency. A further peak can be seen at 54.8Hz corresponding to the second mode of vibration resonance. In the real system, the first peak is unlikely to have an effect, as the operating frequency range of a β-type Stirling engine is 50Hz ± 0.5Hz. The risk occurs towards the second peak, which is close to the operating range.

Referring to Figure 3B, there is shown a frequency response for the simulation, showing the primary mass acceleration magnitude against frequency. The primary mass acceleration is reduced as the frequency increases to reach a minimum at 50Hz.

Referring to Figure 3C, there is shown a frequency response for the simulation, showing the transmissibility magnitude and phase of the primary mass displacement 101 and the absorber system 1 02 against frequency. According to these transmissibility plots, the primary mass displacement 101 at 50Hz shows a dip and the transmissibility plot of the absorber system 102 over the primary mass shows a peak. This indicates that the TMD is tuned to resonate at this frequency in order to minimise the primary mass displacement. This is also clear in terms of the phase plot where at resonance points the phase of transmissibility changes 180 degrees. For this application, the maximum allowed displacement of the primary mass may be restricted to ±20μηι.

Hence, the Bode plot in Figure 3A shows that the primary mass displacement is in the range of few μηι at 50Hz. Moreover, it can be seen that a small drift in the excitation frequency may lead the primary mass displacement to exceed the limit and could cause damage to the system. This clearly points to the fact the current TMD with its passive form would not be a suitable choice when the excitation frequency has a range envelope of [47Hz - 52Hz].

These simulation results were verified against experimental data. Referring to Figure 4, there is shown a plot of acceleration against frequency from experimental results for a number of points on the engine. The close similarity between this graph and the plot of Figure 3B gives confidence in the theoretical model above.

Active Tuned Mass Damper and State Space Model

Referring next to Figure 5A, there is shown a schematic diagram of a known active tuned mass damper (ATMD), using similar features to those of Figure 1 . An

ATMD system is a versatile system that uses an additional forcing function on the mass to perform a broad range of disturbance attenuation by feed forward, feedback or a hybrid form of control. The basis of operation of an ATMD is similar to that of the TMD with an additional force, u(t), added to assist the auxiliary mass via an actuator. Suitable actuators include electric voice coil motors, hydraulic pistons, piezoelectric devices, smart magnetic materials, other electric motors or similar. The force that the actuator will deliver to the structure is dependent on the measurement of the system response. The actuator may be programmed and controlled to deliver a force that opposes the disturbing force and to hold the isolated mass nearly motionless by counteracting the response of the main mass. A feedback element, H(s), provides information about the response of the main mass and feeds it into the actuator control. Specific sensors for the implementation of the ATMD have not been discussed in detail, but these will be well known to the skilled person.

Referring next to Figure 5B, there is shown a schematic diagram of an engine system, particularly comprising a FPSE/LA in accordance with the invention, using an active tuned mass damper. For simulation purposes, it can be understood that the primary system is acted upon by an excitation force F(t), that is assumed to be of constant amplitude and a frequency that is allowed to vary between 45Hz and 55Hz (thereby representing the vibration of the engine. However, this limitation does not need to be true in the theoretical model now presented.

In line with the system shown in Figure 2, this can also be modelled as a 2DoF, continuous, time-invariant system. Applying Newton's Second Law, the equations of motion of such a system are as follows, where U represents the actuator force applied between the engine casing a and the absorber b and F represents the excitation force applied by the engine a.

x =—k x — c x — k ₂ x ₁ — x ₂ ) ^{~~ c} i(*i ^{~~ x'} ₂ )— U + F m ₂ x ₂ = -k ₂ x ₂ - Xi) - c ₂ (x ₂ - x ^' i) + U

These equations can be re-arranged to form a state space model. The vector x = [x _t ; x ₂ ; x ; x ₂ ] defines the system states and another vector u = [F; U] defines the system inputs. Then, the state space equations can be expressed in the following form, where y = x represents the system outputs.

Hence, [C] is a 4x4 identity matrix and [D] is a 4x2 zero matrix.

Referring now to Figure 5C, there is depicted a simplified model of the system of Figure 5B, in accordance with the state space equations. This is a MIMO system 100, represented as a block with two inputs and four outputs, where x ₃ = x _x and x ₄ = x ₂ .

Control Strategies

In the above model, only the actuator force U can be controlled. The excitation force F cannot typically be adjusted or influenced. Nevertheless, feed forward, feedback or a hybrid form of control are all possible. A number of control approaches will now be described, although the skilled person will appreciate that other control strategies may be considered. The control strategies to be discussed are:

1 . feed forward;

2. feed forward with feedback provided by a PID controller; and

3. optimal control using a linear-quadratic approach.

Feed Forward Control Strategy

In this strategy, the actuator force that is placed between the two masses and provides a force u(t) should take the following form.

U = (t) = [a(x - x ₂ ) + /?¾ - x ₂ ) + y x - x ₂ )] The coefficients α, β and γ represent acceleration, velocity, and displacement or position feedback gains respectively.

The acceleration feedback gain a and the position feedback gain γ modify the natural frequency of the absorber as shown below.

The velocity feedback gain β alters the damping of the absorber c ₂ (and these two quantities have the same dimensional units). For an absorber without damping, β = -c ₂ . For a practical point of view, the combination of a and β or β and γ should be considered for optimal control, as it can modify the natural frequency of the absorber mass and its damping coefficient.

For this ideal system ignoring delays, the feedback gains should be set on the basis of the excitation frequency and the following conditions are to be satisfied by the feedback gains to ensure stability of the system. i -m ₁ m ₂

>

m^ + m ₂ )

? > -¾

Y > -k ₂

According to the preceding explanation, it can be viewed that a feedback of a force that is proportional to the relative displacement of the two masses allows altering the stiffness matrix which in turn alters the natural frequency of the absorber. Similarly, a feedback force that is proportional to the relative acceleration of the two masses changes the mass matrix and hence the natural frequency of the TMD. On the other hand, a velocity feedback gain cannot alter the natural frequency of the absorber mass, but it can affect the damping in the absorber mass therefore affecting the bandwidth.

An intention of the invention is to achieve minimal amplitudes of engine vibration within a wider frequency bandwidth. The proposed control system strategy uses acceleration or position feedback gains in order to alter the natural frequency of the absorber mass in addition to a negative velocity feedback that would allow for further attenuation in the vibration of the primary mass.

Referring to Figure 6, there is depicted a model of an engine system and first controller (the feed forward controller discussed above) in accordance with Figure 5B. This is a block diagram representing the feed forward control strategy with position, acceleration, and velocity gains in a simulation model. The calculation of the gains γ and a is performed based on the equations presented as follows, assuming prior knowledge of the excitation frequency.

1 . A sensor is provided that measures one or more of: the relative displacement; the relative velocity; and the relative acceleration of the absorber mass with respect to the primary mass (the engine).

2. The excitation frequency (the frequency of F) should be determined from an electrical parameter (such as the grid frequency) or using the sensor signal.

3. One or both of: the acceleration feedback gain a; and position feedback gain γ are adjusted in order to change the absorber frequency based on the excitation frequency. This will be discussed below. The velocity feedback gain can also be set.

4. Then, the control system can determine the actuator force U to be applied in accordance with one of the following expressions. The first expression relates to acceleration and velocity control (AD) and the second relates to position and velocity control (PD).

(t) = a(x - x ₂ ) + β(χ - x ₂ )

(t) = y{x - x ₂ ) + β{χ - x ₂ ) ln determining position feedback gain, an aim is to derive the formula that relates the gain γ to produce a frequency shift in the absorber system to track the excitation frequency. First of all, it is worth defining ω,ηο as the desired increment in the absorber natural frequency and o as the new absorber natural frequency that is to be altered by the actuator force.

Then, the following can be defined.

By squaring both sides, the value of the gain γ that is required to alter the natural frequency of the absorber system is calculated by using the following formula.

γ = m ₂ (a) _inc + a) _ori g) ² - k ₂

Figure 7 A shows a plot of position feedback gain against frequency increment, based on this formula.

Acceleration feedback gain can similarly be calculated. The equation that relates the acceleration feedback gain a with the increment in frequency can be written as

Figure 7B shows a plot of acceleration feedback gain against frequency increment, based on this formula.

Referring to Figure 6B, there is depicted an alternative model of an engine system and first controller. This uses the block diagram representation of Figure 5C. The controller is feed forward because the feedback gains are calculated from the force excitation frequency, which is an input to the model. Feed forward with Feedback Control Strategy

This strategy is based on the feed forward strategy previously described with the addition of a slow feedback controller, which aims (tends) to minimise the acceleration or displacement of the primary mass (that is, the engine case). This could be achieved by sensing the primary mass displacement, acceleration or both.

Referring to Figure 8, there is illustrated a model of an engine system and a second controller in accordance with Figure 5C. Again, the MIMO 2DoF model of the FPSE/LA with active absorber 100 is shown. The lower part of the diagram shows the feed forward control part 1 10, as discussed above, which is a fast controller. The upper part shows a slow PID controller 120. The respective outputs of both controllers are summed, by summer 125 in order to provide the actuator force input U to the 2DoF model 100.

A reference value 130 is provided. This reference value 130 can relate to acceleration of the engine casing {x ) or its position (x^. In the former case, the switches are set to position P-i and in the latter, the switches are set to position P ₂ . RMS block 140 is used to give a root-mean-squared value for acceleration or position and this is subtracted from the reference value to provide an error signal, which acts as an input to PID controller block 150. The output of the PID block 150 is provided to limiter block 160, which prevents the output from being within 10% of a maximal value. The output of the limiter block 160 is a _P | _D (Pi) or γ _Ρ | ₀ (P ₂ ). This is multiplied by the relative acceleration (Pi) or by the relative position (P ₂ ). The result (u _P | _D ) is added to the feed forward controller output (U _F F) by the summer 125, as discussed above.

When defining (for example) the spring stiffness, damping coefficients and mass elements, components' tolerances and temperature dependencies add uncertainty, which can complicate the control strategy. Fine tuning of the system is desirable. This can be assisted by the addition of the slow PID controller 120, which can finely adjust the acceleration or position gains for minimisation of vibration of the primary mass. The gain provided by the slow PID controller 120 is a small fraction of the maximal gains, as a result of the limiter block 160. By adding this slow PID controller 120 in parallel with the fast feed forward controller 1 10, the approach becomes practical and feasible.

The reference value 130 for the slow PID controller 120 is set as equivalent to no vibration of the primary mass (that is, x - ref = 0 or x _t - ref = 0). The PID block 150 should therefore minimise the error signal, which is its input. Optimal Control

Another control strategy has been proposed based on optimal control with a linear quadratic (LQ) controller with a novel approach based on including the excitation force and the control force as part of the model in the inputs matrix B for the calculation of the controller gains, since the excitation force is known. Referring to Figure 9A, there is depicted a second model of an engine system in accordance with the invention. This is a representation of the state space model of the system in the form of a block diagram..

Optimal control is one of the modern control methods that may be used in the area of vibration reduction. In terms of active vibration control, the optimal control method may achieve the optimal solution by calculating the feedback gains that result from the minimization of a cost function or a performance index based on previous knowledge of the system. In other words, the feedback gains are optimised in order to achieve a control u(t) that minimises the following cost function.

] = ]{%, t, u(t))

Referring to Figure 9B, there is shown the model of Figure 9A with an additional third type of controller, which is an LQ controller. Here, the actuator force u(t) is calculated using the following expression.

u(t) = -[K]x

The vector x is a full state-feedback vector that has all the states and [K] is a constant gain matrix calculated by the LQ controller based on the following.

[K] = -R- ¹ B ^T P

where P is a solution of the nonlinear Algebraic Riccati Equation (ARE) shown below.

A ^T P + PA - PBR- ¹ B ^T P + Q = 0

Matrices Q and R are experimentally determined, using Bryson's Rule as a first approximation.

The cost function for / that is suitable for vibration control can be expressed as follows

with [Q] (n x n) and [ff] (p x p) as positive definite symmetric matrices, n the number of states and p the number of inputs. The scalar quantity x ^T Qx is quadratically related to the outputs of the system under control with x being an n x l matrix. The scalar quantity u ^T Ru is quadratically dependent on the control effort being input by the system with vector u of dimensions p x 1. The following parameters then apply.

9ii 9l2 9l3 9l4 ^"

¾1 922 923 924

[Q] = 931 932 933 934

.94 943 944.

in 1 ^Λ ]

i=l =1

'11 12

^[R] - [r ₂ i r ₂₂ j

U _l = F(tJ

u =

"2 = fit).

Ru VijUjUj

i=l 7=1

It can be seen from this that the diagonal items of Q and R matrices penalise the individual states and the off-diagonal items penalise the combinations of the states.

Referring to Figure 1 0, there is depicted a model of an engine system and the third controller in accordance with the invention, based on the model of Figure 5B.

One of the inputs of the system, excitation force F, is not controllable, though. Only the second input, U (the actuator force) can be controlled to affect the vibration. In view of this, the control strategy can be understood by use of superposition theory. Referring to Figure 1 1 , there is shown a schematic diagram showing the superposition relationship. Superposition can be applied in any linear system that is lumped. The analysed engine and absorber system is a MIMO Linear Time-Invariant (LTI) system and therefore superposition theorem can be applied.

The procedure for this control strategy is then as follows.

1 . R and Q are experimental matrices, determined using Bryson's Rule.

2. The Ricatti equation is solved for P.

3. Once P is known, then K can be calculated using R, B and P using the

expression above.

4. The actuator force (U) is KX(t), which is a single-valued input, as B has the dimensions of one column.

Whilst optimal, this approach may be difficult to operate in practice. A better practical result may be possible using both inputs instead of just one. Then, the B matrix has two columns, instead of one as noted above. For example, the following may be

0

0

B= l/m ₁ —l/m ₁

0 l/m ₂

Referring now to Figure 12, there is depicted the block diagram representation of Figure 9A with the addition of a LQ controller. This is an LQ controller, which again assumes that we have access to the position and acceleration of both the engine casing and the absorber mass (that is, the whole vector x = [x^, x ₂ ; x ; x ₂ ]) . In comparison with Figure 9B, a precompensation block N is additionally provided. This deals with a tracking problem, which implies that the output y will follow the reference input r.

It is not always possible to have access to all of the states, however. Then, a state space estimator (observer) can be used to retrieve the system states from the output vector y. This type of controller is termed as Linear-Quadratic Gaussian (LQG) controller. Referring to Figure 13, there is depicted the block diagram representation of Figure 9A with the addition of a LQG controller. This is basically a combination of the LQ controller with a state space estimator (termed an LQE), such as a Kalman Filter. For example, this may be able to calculate the system states from { - x ₂ ), (xi - x ₂ ) and (x^ - x ₂ ).

The state-feedback control law results in a closed-loop system of the following form.

The coefficients of [Q] and [R] represent the weighting of one state or input versus another. Initially a first choice can be based on Bryson's rule which estimates the coefficients of the LQ matrices by taking one over the square of the maximum range of the associated state. Nevertheless, the ultimate choice of the coefficients is determined by trial and error, as noted above.

Typical values of [Q], [R]and [K] are presented in the following table. States Matrix Q Control Matrix Gain Matrix K

R

1 .0( 3+08 * 1 .0e-03 * 1 .0e+04 * rO.001 0 ] Γ 2145 87.1 2233.8 0.0125]

L o 0.5J L—4.36 -0.0868 -4.5 0.012 J

Since the gain matrix [K] is a 2 x 4 matrix, upon multiplying it with the full 4 x 1 state vector x, the resulting vector is a 2 x 1 vector with elements corresponding to the excitation force and control force respectively. Since the aim is not necessarily to track or reduce the excitation force, only the element associated with the control force ^" (£) is considered, while ignoring the first element related to the excitation force. The stability of the system with the state-space model shown above was examined by checking the eigenvalues of matrix [A] which proved stability.

New grid connection/disconnection regulations for small scale generators have imposed a huge design constraint in FPSE/LA generators when the machine is grid connected. Currently, this type of machine is cost effective when it operates in a narrow bandwidth of frequencies. The inclusion of active control strategies is an option to explore when aiming to meet new regulations. As a result, the conversion of the passive absorber that already exists in the β-type Stirling engine has been converted into an active TMD. As discussed above, the response of the simulated model showed a great match with the experimental data that have been recorded. This allows the assumption that the theoretical model represents the working system well. A feed forward and optimal LQ control strategies have been investigated from a frequency and time domain perspectives. Results have shown that both strategies have succeeded to minimise the vibration of the Stirling engine case to settle within the specified limit of peak 20μηι over a wide bandwidth of 1 0Hz with some little differences in terms of amplitudes, phase shifts, and settling times.

Although specific embodiments have now been described, the skilled person would understand that various modifications and variations are possible. For instance, other models for the same engine design or different engine designs may produce an alternative control strategy, which may be based on the same variables or slightly different variables than those discussed above. It may be possible to use active damping technologies other than a tuned mass damper, for example. Other issues may also be considered, such as the effect of delays in the system and the type of actuators that are suitable for performing the active damping in this particular application within the required specifications. The damper system could use single or multiple coil springs arranged symmetrically or asymmetrically or even incorporate flat planar springs.

Moreover, different forms of control may be possible from those described. In particular, a hybrid feed forward with feedback controller may use an alternative to a PID controller to provide feedback. Any control strategy that minimises the error might be used as an alternative to PID block 150, for instance.