Patent Description:
The present disclosure relates generally to systems and methods for controlling the temperature of a device, such as a semiconductor electronic device under test.

Systems for testing and handling electronic devices, such as packaged integrated circuit chips and unpackaged, bare "chips" or other devices under test (DUT), conventionally may include a temperature control system to maintain the temperature of the electronic device near a constant set point temperature while the device is being tested. Any type of circuitry can be integrated into the DUTs, such as digital logic circuitry or memory circuitry or analog circuitry. Also, the circuitry in the DUT can be comprised of any type of transistors, such as field effect transistors or bi-polar transistors.

It is desirable to keep the temperature of a device constant while it is tested. For example, it is common to test various integrated circuits for failure at certain temperatures. Further, a common practice in the chip industry is to mass produce a particular type of chip, and then speed sort them and sell the faster operating chips at a higher price. CMOS memory chips and CMOS microprocessor chips are processed in this fashion. However, because the speed with which the chip operates may be temperature dependent, the temperature of each chip must be kept nearly constant while the speed test is performed in order to determine the speed of such chips properly.

One challenge in keeping the temperature of a chip or collection of chips constant during testing is minimizing a temperature gradient across the one or more chips being tested. For example, a block (e.g., plate, holder, nest, etc.) may be placed in thermal contact with one or more chips, while the temperature of the block is controlled to attempt to keep the temperature of the one or more chips constant during testing. However, particularly for relatively large blocks, the temperature at one area of the block may become different than at another area of the block. In such a case, the resulting temperature gradient across the block may result in uneven temperatures across the one or more chips being tested, potentially compromising test results. Accordingly, systems and methods for minimizing temperature gradients across one or more chips during testing are needed. <CIT> discloses a thermal system comprising: a plurality of thermal elements; a control system having at least three power nodes, wherein a thermal element of the plurality of thermal elements is connected between each pair of power nodes.

One implementation of the present disclosure is a system for handling integrated circuit devices. The system includes a block comprising a plurality of thermally-coupled zones and a plurality of heaters. Each of the plurality of heaters is controllable to provide heat to one of the plurality of thermally-coupled zones. The system also includes a plurality of temperature sensors. Each of the plurality of temperature sensors configured to measure temperature of one of the plurality of thermally-coupled zones. The system also includes a control circuit configured to receive, from the plurality of temperature sensors, a temperature measurement for each of the plurality of thermally-coupled zones, collect the temperature measurements in a temperature vector in a real coordinate system, and transform the temperature vector to a normal coordinate system. The normal coordinate system provides a plurality of uncoupled equations. The control circuit is also configured to determine, based on the plurality of uncoupled equations and a desired temperature gradient across the plurality of thermally-coupled zones, a desired power vector in the normal coordinate system. The control circuit is also configured to transform the desired power vector in the normal coordinate system to the real coordinate system to generate a power vector and control the plurality of heaters in accordance with the power vector to substantially achieve the desired temperature gradient across the plurality of thermally-coupled zones.

Referring generally to the drawings, systems and methods for providing temperature control of a thermal system with multiple thermally-coupled zones are shown, according to exemplary embodiments. The systems and methods described herein may be implemented in a variety of testing and handling system for devices, such as integrated circuit devices (e.g., devices under test or DUTs). In such systems, it may be advantageous to substantially maintain the DUTs at a desired temperature during testing. One approach is to place the DUTs in thermal contact with a block or other structure and control the temperature of the block. The DUTs might also be arranged in multiple sockets. In the exemplary systems contemplated herein, the DUTs are positioned relative to each other such that a significant temperature gradient may exist across the DUTs,, e.g., , a temperature at a first point on the block and a temperature at a second point on the block may be sufficiently different to compromise test results.

The present disclosure introduces the concept that the DUTS are positioned in multiple thermally-coupled zones, where each zone may have a different temperature. As described in detail herein, each zone may be provided with an independently-controllable thermal control device to facilitate better management of the temperature gradient across the block. The thermal control device may include, for example, a heater, although other thermal control devices may be used to heat or cool the one of multiple zones. Because the zones are thermally coupled to one another, the operation of each heater affects the temperature of all of the zones, and, therefore, the amount of heat required from each of the other heaters. In some cases, a fan is provided at each zone to facilitate removal of heat from the zone. Systems and methods for controlling the heaters and/or fans that account for thermal coupling between zones, as described in detail below, may therefore be advantageous in maintaining the block at a desired temperature across multiple zones and/or providing some other desired temperature gradient across the multiple zones.

Referring now to <FIG>, a thermal system <NUM> is shown, according to an exemplary embodiment. The thermal system <NUM> is shown to include a block <NUM> having four thermally-coupled zones, namely zone A <NUM>, zone B <NUM>, zone C <NUM>, and zone D <NUM>. It should be understood that the thermal system <NUM> is one of many possible thermal systems contemplated by the present disclosure and that reference is made to thermal system <NUM> for the sake of example. For example, in various embodiments, various numbers of zones may be included (e.g., three zones, five zones, six zones, etc.). The block <NUM> may be made of a conductive metal (e.g., copper) or other suitable thermal material. A top surface <NUM> of the block <NUM> may be configured to hold (carry, support, retain) one or more integrated circuit devices (e.g., chips) for testing.

As shown in <FIG>, the block <NUM> is coupled to a heat exchanger <NUM>. The heat exchanger <NUM> may include fins and/or other structure(s) for facilitating heat transfer from the block <NUM>. The heat exchanger <NUM> may be found at a substantially constant ambient temperature Ta. In the embodiments described herein, the ambient temperature Ta is considered to be unaffected by the thermal system <NUM> (i.e., heat may be transferred from the thermal system <NUM> to the environment without altering Ta).

The thermal system <NUM> is further shown to include multiple heaters (e.g., four heaters), multiple temperature sensors (e.g., four temperature sensors), and multiple fans (e.g., four fans). Each zone may be aligned with one heater, one temperature sensor, and one fan. In the example shown, zone A <NUM> includes temperature sensor A <NUM>, heater A <NUM>, and fan A <NUM>; zone B <NUM> includes temperature sensor B <NUM>, heater B <NUM>, and fan B <NUM>; zone C <NUM> includes temperature sensor C <NUM>, heater C <NUM>, and fan C <NUM>; and zone D <NUM> includes temperature sensor D <NUM>, heater D <NUM>, and fan D <NUM>.

Each temperature sensor <NUM>-<NUM> is configured to measure the temperature of the block <NUM> at the corresponding zone. That is, temperature sensor A <NUM> measures the temperature at zone A <NUM> ("TA"), temperature sensor B <NUM> measures the temperature at zone B <NUM> ("TB"), temperature sensor C <NUM> measures the temperature at zone C <NUM> ("TC"), and temperature sensor D <NUM> measures the temperature at zone D <NUM> ("TD"). In some embodiments, the temperature sensors <NUM>-<NUM> include resistance thermal detectors (RTDs). The temperature sensors <NUM>-<NUM> may provide analog and/or digital indications of temperature to a control circuit <NUM>, shown in <FIG> and described in detail below. The temperature sensors <NUM>-<NUM> are located at or near a top surface <NUM> of the block <NUM>, such that the temperature sensor <NUM>-<NUM> measure temperature proximate a position of one or more chips positioned on the block <NUM> for testing.

Each heater <NUM>-<NUM> is configured to provide heat to the block <NUM> at the corresponding zone. For example, heater A <NUM> provides heat to the block <NUM> at zone A <NUM>. Each heater <NUM>-<NUM> may be operable at variable power to provide a variable amount or rate of heat to the block <NUM>. As described in detail below, the heaters <NUM>-<NUM> are independently controllable, such that each heater A <NUM> may operate at a different power than heater B <NUM>, heater B <NUM> may operate at a different power than heater C <NUM>, and so on. The heaters <NUM>-<NUM> may convert electricity into heat through electrical resistance and/or generate heat in some other manner.

Each fan <NUM>-<NUM> is configured to blow air across the block <NUM> at the corresponding zone to facilitate the transfer of heat out of the block <NUM> via the heat exchanger <NUM>, for example to draw the temperature of the block <NUM> towards the ambient air temperature. The fans <NUM>-<NUM> may be independently controllable to operate at various powers (e.g., fan blade speeds, airflow levels, etc.).

Referring now to <FIG>, a block diagram of the thermal system <NUM> of <FIG> with a control circuit <NUM> is shown, according to an exemplary embodiment. The control circuit <NUM> is communicably coupled to the temperature sensor A <NUM>, temperature sensor B <NUM>, temperature sensor C <NUM>, and temperature sensor D <NUM>. In various other embodiments, the control circuit <NUM> may be communicably coupled to any number of temperature sensors. The control circuit <NUM> can receive temperature measurements from the temperature sensors. More particularly, in the example shown, the control circuit <NUM> can receive the temperature TA at zone A <NUM> from temperature sensor A <NUM>, the temperature TB at zone B <NUM> from temperature sensor B <NUM>, the temperature TC at zone C <NUM> from temperature sensor C <NUM>, and the temperature TD at zone D <NUM> from the temperature sensor D <NUM>.

The control circuit <NUM> is configured to control the heaters <NUM>-<NUM> and/or the fans <NUM>-<NUM> based on the temperature measurements TA, TB, TC, TD. In the example shown, the control circuit <NUM> controls heater A <NUM> to operate at power PA, heater B <NUM> to operate at power PB, heater C <NUM> to operate a power PC, and heater D <NUM> to operate at power PD. In the example shown, the control circuit <NUM> may also control fan A <NUM> to operate at power PFanA, fan B <NUM> to operate at power PFan B, fan C <NUM> to operate at power PFanC, and fan D <NUM> to operate at power PFanD. The control circuit <NUM> may thereby control the amount of heat added and/or removed to each of the thermally-coupled zones <NUM>-<NUM>. To determine the values of P = {PA, PB, PC, PD, PFanA, PFan B, PFαnC, PFanD}, the control circuit <NUM> may follow process <NUM> shown in <FIG> and described in detail with reference thereto below. It should be understood that in various other embodiments (e.g., with a different number of zones), the control circuit <NUM> may receive various numbers of temperature measurements and output various number of power levels to control various numbers of heaters and/or fans.

The control circuit <NUM> is configured to generate the power controls Pi based on the temperature measurements Ti to control the temperature and temperature gradient across the thermally-coupled zones to achieve a desired temperature gradient (e.g., zero gradient). In other words, in the formulation contemplated by the present disclosure, the control circuit <NUM> is configured to manage a dynamically-coupled thermal system that may be approximated by a multi-degree-of-freedom discrete system of order n defined by the system equation C Ṫ + K T = P. In this formulation, C is the thermal capacity (or thermal mass) matrix with units of J/°C. K is the thermal conductivity matrix with units of W/°C. P is the power vector with units of W (e.g., a vector of PA through PD). T is the temperature vector (e.g., a vector of TA through TD). Ṫ is the time-derivative of the temperature vector T (i.e., Ṫ = δT /δt, where t is time) with units of °C/s. Under a static analysis (i.e., Ṫ = <NUM>), the power vector can be calculated based on a given temperature vector as K T = P. Inversely, in the static case the temperature can be calculated based on a given power vector as T = K-<NUM> P.

In this formulation, the vectors (indicated by a single underline) are of length n and the matrices are n-by-n square. The parameter n may be any integer and may equal a count of the number of thermally-coupled zones and/or the number of controllable devices (i.e., heaters and/or fans). For example, in the example thermal system <NUM> of <FIG> having four zones <NUM>-<NUM> and four heaters <NUM>-<NUM>, n may be four. In some cases, to account for the four fans <NUM>-<NUM>, n may be eight.

In the dynamic, thermally-coupled system managed by the control circuit <NUM>, the thermal coupling between various zones prevents conventional control approaches from being directly applied to the system equation provided above. Accordingly, as described in detail with reference to <FIG>, the control circuit <NUM> may generate control signals by transforming from a real coordinate system to a normal coordinate system to generate a set of uncoupled equations in the normal coordinate system, utilize the uncoupled equations to determine control inputs in the normal coordinate system, and convert back into the real coordinate system to determine the power inputs for use in controlling the heaters <NUM>-<NUM> and/or fans <NUM>-<NUM>. The following paragraphs provide a derivation of the mathematical basis for this approach.

To address the system equation C Ṫ + K T = P, the homogenous equation C Ṫ + K T = <NUM> may be solved using an eigenvalue approach. Assume T = a e-λt , which implies Ṫ = λ T and [K - λC ] T = <NUM>. Non-trivial solutions (i.e., T ≠ <NUM>) require det |K - λC| = <NUM>. There are n solutions, namely eigenvalues λi, i = <NUM>,. Each eigenvalue λi has a corresponding eigenvector Ei that is scaled such that <MAT>, where <MAT> is the transpose of the vector Ei. Orthogonality properties provide that <MAT>; and <MAT>.

In a particular case, various methods may be used to find the numerical values of the eigenvalues λi and eigenvectors Ei for a physical thermal system. Using the eigenvectors Ei, a modal matrix V may be defined V = [E<NUM>, E<NUM>,. VT denotes the transpose of V. A vector θ may then be defined as T = V θ, where θ is an n-length vector of elements θi. θ may be characterized as a representation of the temperature vector T in a normal coordinate system (whereas T defines temperature values in a real coordinate system). Given these definitions, the system equation C Ṫ + K T = P can be rewritten as VTC V θ̇ + VTK V θ = VT P = F, where F is a vector of elements Fi that may be characterized as a representation of the power vector P in the normal coordinate system (whereas P defines power values in a real coordinate system).

The orthogonality principles above imply that V T C V = I, where I is the identity matrix, and V T K V = Λ, where Λ is a diagonal matrix containing the eigenvalues λi. Accordingly, the system equation V T C V θ̇ + V T K V θ = F can be reduced to θ̇ + Λ θ = F, or, equivalently, n uncoupled equations of the form θ̇i + λiθi = Ei T P = Fi. The approach outlined in the preceding paragraphs thereby transforms a coupled system of equations that describes the dynamically-coupled thermal system in a real coordinate system to a collection of uncoupled equations in a normal coordinate system.

In online control, the control circuit <NUM> may calculate the values of θi from measured (real-coordinate) temperature values Ti based on: T = V θ ⇒ θ = V-<NUM>T = [VT C ]T. One or more of a variety of known control approaches may then be applied using the values of θi to generate values of Fi in the normal coordinate system based on the uncoupled equations θ̇i + λiθi = Fi. The control circuit <NUM> may then convert Fi back to the real coordinate system to determine values of Pi following: F = VT P ⇒ P = (VT)-<NUM>F = [ C V ] F, which can then be directly used to control the heaters <NUM>-<NUM> and/or fans <NUM>-<NUM>.

Referring now to <FIG>, a flowchart of a process <NUM> for controlling the heaters <NUM>-<NUM> and/or fans <NUM>-<NUM> is shown, according to an exemplary embodiment. <FIG> provide example illustrations useful for describing the process <NUM>, and reference is made thereto in the following discussion. Process <NUM> can be executed by the control circuit <NUM> of <FIG>.

At step <NUM>, the control circuit <NUM> identifies the matrices C and V to be used in converting between the real coordinate system and the normal coordinate system. More particularly, as θ = [VT C ]T and P =[ C V ] F, the control circuit <NUM> may identify numerical values for each element of [VT C ] and [ C V ] at step <NUM>. In some cases, the values may be automatically derived by the control circuit <NUM>. In other cases, the values may be derived be an engineer and pre-programmed on the control circuit <NUM>.

For example, the values of the elements of matrices C and V may derived analytically based on a circuit-style diagram, for example diagram <NUM> of <FIG>. Diagram <NUM> of <FIG> corresponds to the thermal system <NUM> of <FIG>. As shown in diagram <NUM>, zone A <NUM> is at temperature TA, has thermal capacity CA, and receives power PA. Zone A <NUM> is shown as thermally coupled to the ambient air (temperature Ta) across a resistance RA-a, as well as to zone B <NUM> across a resistance RA-B. Zone B <NUM> is at temperature TB, has thermal capacity CB, and receives power PB. Zone B <NUM> is shown as thermally coupled to the ambient air (temperature Ta) across a resistance RB-a, as well as to zone A <NUM> across a resistance RA-B and zone C <NUM> across resistance RB-C. Zone C <NUM> is at temperature TC, has thermal capacity Cc, and receives power PC. Zone C <NUM> is shown as thermally coupled to the ambient air (temperature Ta) across a resistance RC-a, as well as to zone B <NUM> across a resistance RB-C and zone D <NUM> across resistance RC-D. Zone D <NUM> is at temperature TD, has thermal capacity CD, and receives power PD. Zone D <NUM> is shown as thermally coupled to the ambient air (temperature TD) across a resistance RD-a, as well as to zone C <NUM> across a resistance RC-D.

From the diagram <NUM>, the thermal capacity and conductivity matrix can be defined as: <MAT>. In the example of <FIG>, <MAT>, where Cp is the specific heat [J/kg] of the material of the block <NUM> and m is the mass of the block <NUM>.

Furthermore, from the diagram <NUM> of <FIG>, the thermal conductivity matrix can be defined as: <MAT>.

In the example of <FIG>, RA-a = RB-a = RC-a = RD-a and RA-B = RB-C = <MAT>, where l is the length of the block <NUM>, Area is the cross-sectional area of the block <NUM>, and k is the thermal conductivity of the block material.

From those definitions based on the diagram <NUM> of <FIG>, numerical values can be calculated for each element of the matrices C and K. The matrix C may thereby be identified at step <NUM>. To determine the values of the elements of V, the actual values for C and K may be plugged into the system equation C Ṫ + K T = P. Given a set of example values associated with the diagram <NUM> of <FIG>, this may yield: <MAT>.

From this statement of the system equation, eigenvalues and eigenvectors can be found for use in building the matrix V for step <NUM> of <FIG>. The eigenvectors are collected as the columns of V, yielding, for this example, V = <MAT>. These eigenvectors are plotted on graph <NUM> of <FIG>. In various alternative examples having various other values for thermal conductivity, thermal capacity, mass, etc. the eigenvalues and eigenvectors have different values. For example, <FIG> shows a graph <NUM> of the eigenvectors for an example in which CA = <MAT>. In such an example, the increased thermal capacity of zone B <NUM> creates an asymmetry reflected in the matrix V (and visible on the graph <NUM>), but which may not otherwise affect the analysis presented herein.

Still referring to <FIG> and step <NUM> of process <NUM>, the matrices C and V may thereby be identified for use by the control circuit <NUM> in online control. In some embodiments, the control circuit <NUM> directly identifies the matrices [VT C ] and [ C V ] at step <NUM>. For the sake of completeness of the foregoing example, in the example laid out above these matrices may be identified as: <MAT> with [ C V ] equal to the transpose of [VT C ].

Still referring to <FIG>, at step <NUM> the temperature sensors <NUM>-<NUM> collect temperature measurements at each of the zones <NUM>-<NUM> and provide the temperature measurements (e.g., TA,. , TD) to the control circuit <NUM>. The temperature measurements are received by the control circuit <NUM> in a real coordinate system, for example in units of degrees Celsius.

At step <NUM>, a vector T of the real temperature measurements (in the real coordinate system) is converted to the normal coordinate system to determine the feedback θi. That is, θ is calculated as θ = [VT C ]T, with θ made up of entries θi. This conversion allows the real temperature measurements from the temperature sensors <NUM>-<NUM> to be used as feedback in the normal coordinate system. At step <NUM>, the values θi in the normal coordinate system are applied to uncoupled equations θ̇i + λiθi = Fi to form scalar, first-order differential equations.

At step <NUM>, the uncoupled are equations are used to determine desired values of Fi, which may be described as desired power in the normal coordinate system. Various optimizations, control approaches, etc. may be applied in various situations. For example, desired values of Fi may be determined based on a temperature setpoint for the block <NUM>, to minimize a temperature gradient across the block <NUM> (e.g., to minimize the difference in the temperature measurements TA, TB, Tc, TD), and/or to optimize an economic cost of operating the heaters <NUM>-<NUM>. The values Fi are collected in the vector F.

At step <NUM>, the power vector F in the normal coordinate system is converted to the real coordinate system to determine a real-coordinate power vector P. That is, the control circuit <NUM> calculates P as P =[ C V ] F. The control circuit <NUM> thereby calculates P, which is made up of the real-coordinate powers PA through PD in the example shown in <FIG>. At step <NUM>, the control circuit controls the heaters <NUM>-<NUM> and/or fans <NUM>-<NUM> to achieve the power indicated by P. For example, as shown in <FIG>, the control circuit <NUM> may transmit a control signal indicating the power PA to heater A <NUM>, a control signal indicating the power PB to heater B <NUM>, a control signal indicating the power PC to heater C <NUM>, and a control signal indicating the power PD to heater D <NUM>. The control circuit <NUM> may thereby overcome the challenges associated with thermal coupling between the zones <NUM>-<NUM>.

In some alternative embodiments, an eight-channel control system may be used to provide for control of the four heaters <NUM>-<NUM> as well as the four fans <NUM>-<NUM>. In some embodiments, the control of the four fans <NUM>-<NUM> may be assumed to be uncoupled with the feedback for each fan being the heater power, which, in this example, may be measured and provided to the control circuit <NUM>. In such an example, an error signal may be defined as: Er = <MAT>, where Tspi is a temperature setpoint and Pspi is a heater power setpoint with (i = A, B, C, D) and the bottom four rows correspond to the control of the fans <NUM>-<NUM>. As derived above, converting to the normal coordinate system requires multiplication by [VT C ] to get θ = [VT C ]Er. The matrix [VT C ] reflects the fact that the bottom four rows are uncoupled. Using the example values calculated in the example above, the eight-channel version of [VT C ] may be defined as <MAT> so that the uncoupled (bottom four) elements of Er (corresponding to the fans <NUM>-<NUM>) are unaffected by the coordinate transformation. Note that[C V] has a similar form, such that the uncoupled elements are also unaffected by the coordinate transformation back to real coordinates under P =[ C V ] F. This is true even where the rows/elements such that the uncoupled degrees of freedom are interspersed among the coupled degrees of freedom (e.g., placed in positions <NUM>, <NUM>, <NUM>, and <NUM> in Er).

In other embodiments, the control circuit <NUM> may use the output of some channels as the input into other channels. For example, in some cases the estimated heater power from one of the channels may be used as feedback for a cooling source such as a fan (e.g., one of fans <NUM>-<NUM>), which may allow for minimization of heater output by controlling fan rotation. In such a case, using the Er vector defined above, heater outputs from the first four rows may be used as inputs to changes <NUM>, <NUM>, <NUM>, and <NUM>, respectively. The normal coordinates can then be defined as <MAT>.

In other cases, the matrix B may be used to provide a fixed output (e.g., fixed PWM). In such a case, in the example above the matrix B may be defined such that [B] = <MAT>.

It should be understood that many such variations and formulations are contemplated by the present disclosure to account for various features of the thermal system managed by the control circuit <NUM> and/or to facilitate optimization of various desired parameters.

Although the figures show a specific order of method steps, the order of the steps may differ from what is depicted. Also two or more steps can be performed concurrently or with partial concurrence. Such variation will depend on the software and hardware systems chosen and on designer choice. All such variations are within the scope of the disclosure. Likewise, software implementations could be accomplished with standard programming techniques with rule based logic and other logic to accomplish the various connection steps, calculation steps, processing steps, comparison steps, and decision steps.

The construction and arrangement of the systems and methods as shown in the various exemplary embodiments are illustrative only. Although only a few embodiments have been described in detail in this disclosure, many modifications are possible (e.g., variations in sizes, dimensions, structures, shapes and proportions of the various elements, values of parameters, mounting arrangements, use of materials, colors, orientations, etc.). For example, the position of elements can be reversed or otherwise varied and the nature or number of discrete elements or positions can be altered or varied. Accordingly, all such modifications are intended to be included within the scope of the present disclosure. The order or sequence of any process or method steps can be varied or re-sequenced according to alternative embodiments.

As used herein, the term "control circuit" may include hardware structured to execute the functions described herein. In some embodiments, the "control circuit" may include machine-readable media for configuring the hardware to execute the functions described herein. The control circuit may be embodied as one or more circuitry components including, but not limited to, processing circuitry, network interfaces, peripheral devices, input devices, output devices, sensors, etc. In some embodiments, the control circuit may take the form of one or more analog circuits, electronic circuits (e.g., integrated circuits (IC), discrete circuits, system on a chip (SOCs) circuits, etc.), telecommunication circuits, hybrid circuits, and any other type of "circuit. " In this regard, the "control circuit" may include any type of component for accomplishing or facilitating achievement of the operations described herein. For example, a control circuit as described herein may include one or more transistors, logic gates (e.g., NAND, AND, NOR, OR, XOR, NOT, XNOR, etc.), resistors, multiplexers, registers, capacitors, inductors, diodes, wiring, and so on).

Claim 1:
A system (<NUM>) for handling integrated circuit devices, comprising:
a block (<NUM>) comprising a plurality of thermally-coupled zones (<NUM>, <NUM>, <NUM>, <NUM>);
a plurality of heaters (<NUM>, <NUM>, <NUM>, <NUM>), each of the plurality of heaters (<NUM>, <NUM>, <NUM>, <NUM>) controllable to provide heat to one of the plurality of thermally-coupled zones (<NUM>, <NUM>, <NUM>, <NUM>);
a plurality of temperature sensors (<NUM>, <NUM>, <NUM>, <NUM>), each of the plurality of temperature sensors (<NUM>, <NUM>, <NUM>, <NUM>) configured to measure temperature of one of the plurality of thermally-coupled zones (<NUM>, <NUM>, <NUM>, <NUM>);
characterized in that the system further comprises a control circuit (<NUM>) configured to:
receive, from the plurality of temperature sensors (<NUM>, <NUM>, <NUM>, <NUM>), a temperature measurement for each of the plurality of thermally-coupled zones (<NUM>, <NUM>, <NUM>, <NUM>);
collect the temperature measurements in a temperature vector in a real coordinate system;
transform the temperature vector to a normal coordinate system, the normal coordinate system providing a plurality of uncoupled equations;
determine, based on the plurality of uncoupled equations and a desired temperature gradient across the plurality of thermally-coupled zones, a desired power vector in the normal coordinate system;
transform the desired power vector in the normal coordinate system to the real coordinate system to generate a power vector; and
control the plurality of heaters (<NUM>, <NUM>, <NUM>, <NUM>) in accordance with the power vector to achieve the desired temperature gradient across the plurality of thermally-coupled zones (<NUM>, <NUM>, <NUM>, <NUM>).