CN106970611A - Network control system sampling period optimal control method - Google Patents

Abstract

The invention provides a kind of network control system sampling period optimal control method, including measurement output variable to controlled device, sampled by tune output variable, and build discrete system model；Set up sampling period T_sWith the relation between network delay τ；Dynamic Output Feedback robust controller is designed, and builds closed-loop model；Whether the closed-loop model obtained by detection meets robust asymptotic stability requirement；Set up sampling period T_sWith the relation between optimal robustness performance γ；Sampling step length h is set, and sets a sampling period T_s'=T_s+ h, repeats said process, stops sampling when closed-loop system loses robust asymptotic stability, defines next sampling period T corresponding to minimum optimal robustness performance γ_s' it is the optional sampling cycle, and it is designated as T_γ.Present invention mainly solves the problem of sampling period optimal control, devise a kind of state output feedback robust controller, it is ensured that system is in the case of extraneous disturbance, still with good stability, dynamic and robustness.

Description

Network control system sampling period optimal control method

Technical field

It it is a kind of sampling week of network control system the present invention relates to network control system automatic control technology field Phase optimal control method.

Background technology

With automative control theory, computer technology, the network communications technology growing and Cross slot interference, control system The structure of system becomes increasingly complex, and spatial distribution is more and more wider, network control system (Networked Control System, NCS) it is a closed-loop feedback control system being made up of real-time network.In network control system, network is situated between as transmission Matter, realizes the data transfer between sensor, controller, actuator and other network nodes, thus realize resource-sharing, it is remote Journey is detected and controlled.Network control system with its cost it is low, connection flexibly, be easily installed extension, be easy to safeguard the advantages of from The limitation of traditional " point-to-point " formula signal control is fundamentally breached, is the objective demand of tele-control system, extensively Applied to fields such as space flight and aviation, intelligent grid, remote fault diagnosis, robot controls.Fig. 2 is the allusion quotation of network control system Type structure, the characteristics of due to network to communication media time-sharing multiplex, when multiple nodes are carried out data transmission by network, usually There are the phenomenons such as information occlusion, disconnecting, multiframe transmission, thus inevitably the delay of information occur, therefore time lag is asked Topic is one of subject matter that network control system faces, and it is often to cause the major reason of system deterioration.On the other hand, With the continuous expansion of modern control system scale, complexity is continuously increased, and external environment condition unpredictability, external disturbance is random Property etc., people is hardly resulted in the description of system determination, therefore consider the external disturbance of system, planned network networked control systems Robust stabili, to remain to keep preferable performance right and wrong when ensureing that the dynamic characteristic of system changes in certain range of disturbance It is often important and necessary.

In actual NCS, due to the limitation of environmental factor or economic condition, network delay is not only existed in system, and And it is difficult to whole states for detecting controlled device, therefore, the research of network control system problem turns into one of focus.Such as Zhu Qixin, Lu Kaihong, Zhu Yonghong, et al. in document《Robust state feedback control of the multi tate without networked control systems with time-delay [J]》The sample frequency of (Suzhou Institute of Science and Technology journal (natural science edition), 2017,34 (1)) middle finger egress is more than one Network control system is referred to as multi tate network control system, and this article utilizes robust control and linear inequality method, devised many Robust State Feedback Controllerss of the speed without networked control systems with time-delay, designed controller can make corresponding closed-loop system Asymptotically stability.The method has the following disadvantages：

1) proposed in text to sensor node, controller node, sliding-model control is carried out with the different sampling periods, to every The sampling period of individual node is still to provide in advance, the robust controller that the method is obtained, and is only under the sampling period Optimum control, belongs to local optimum, does not ensure that the global optimum of system, i.e., under the optimal sampling period, system reaches Optimal robust control；

2) work of author is not consider ideally carrying out for network delay, and in actual industrial system In, network delay is certainly existed；

3) controller designed in text is state feedback robust controller, and in actual engineer applied, the shape of system State is often to be difficult to measurement collection, and this just limits application of this method in engineering.

The content of the invention

The technical problem to be solved in the present invention is for all with fixed sampling present in existing network control technology Phase obtains the problem of robust controller does not ensure that system robustness energy global optimum, and network latency problems, external disturbance is asked There is provided a kind of sampling period optimal control method of network control system for topic.

To achieve the above object, present invention employs following technical scheme.

A kind of network control system sampling period optimal control method, including output is measured to controlled device

Variable and the sampling by tune output variable, its key step are as follows：

Step 1, the measurement output variable to controlled device, sampled by tune output variable, and build discrete system mould Type；

Wherein,

State variable when x (k) is controlled device current time k；

X (k+1) is the state variable at controlled device next controlling cycle (k+1) moment；

Control input amount when u (k) is controlled device current time k；

Control input amount when u (k-1) is previous controlling cycle (k-1)；

Measurement output variable when y (k) is current time k；

Z (k) be current time k when by tune output variable；

External disturbance when w (k) is current time k；

A is controlled device continuous time state variable x (t) coefficient and is a constant, and t is consecutive hours Between variable, T_sFor the sampling period；

B₁For controlled device continuous time control input amount u (t) coefficient and be a constant, τ is network delay；

B₂Coefficient 1 for external disturbance and be a constant；

C₁Coefficient 1 for state variable x (k) and be a constant；

C₂Coefficient 2 for state variable x (k) and be a constant；

D₁Coefficient 2 for external disturbance and be a constant；

D₂Coefficient 3 for external disturbance and be a constant；

Step 2, set up sampling period T_sWith the relation between network delay τ；

Wherein, T_sendTo send cycle, and T_s≥T_send, m is the length of queue, and n is of contained packet in queue Number；

Step 3, design Dynamic Output Feedback robust controller, and build closed-loop model；

The expression formula of the closed-loop model is as follows：

Wherein,

X (k-1) is the state variable at controlled device previous controlling cycle (k-1) moment；

x_c(k) state variable when for controller current time k；

x_c(k+1) it is the state variable at controller next controlling cycle (k+1) moment；

x_c(k-1) it is the state variable at controller previous controlling cycle (k-1) moment；

W (k-1) is the external disturbance at previous controlling cycle (k-1) moment；

A_cFor controller state variable x_c(k) coefficient 1 and be a constant；

B_cTo measure output y (k) coefficient 1 and being a constant；

C_cFor controller state variable x_c(k) coefficient 2 and be a constant；

D_cTo measure output y (k) coefficient 2 and being a constant；

Whether whether the closed-loop model obtained by step 4, detecting step 3 meets robust asymptotic stability requirement, i.e., full Sufficient inequality (4), if meeting inequality (4), closed-loop model meets robust asymptotic stability requirement, into step 5, otherwise Return to step 1；

Wherein,

W₅₁=A_d+B₁₀D_cC₂,W₅₁ ^TFor W₅₁Transposition；

W₅₂=B₁₀C_c,W₅₂ ^TFor W₅₂Transposition；

W₅₃=B₁₁D_cC₂,W₅₃ ^TFor W₅₃Transposition；

W₅₄=B₁₁C_c,W₅₄ ^TFor W₅₄Transposition；

W₆₁=B_CC₂,W₆₁ ^TFor W₆₁Transposition；

For A_cTransposition；

P is the coefficient 1 of liapunov function and is a constant, P^-1For the inverse of P；

S is the coefficient 2 of liapunov function and is a constant, S^-1For the inverse of S；

R is the coefficient 3 of liapunov function and is a constant；

Z is the coefficient 4 of liapunov function and is a constant；

Step 5, set up sampling period T_sWith the relation between optimal robustness performance γ；

Wherein, W₇₅=B₁₀D_cC₂+B₂,W₇₅ ^TFor W₇₅Transposition；For B_cTransposition；γ is optimal robustness performance；ε is mark Amount；For unit matrix；

Step 6, setting sampling step length h, and set a sampling period T_s'=T_s+ h, repeat step 1,2,3,4,5, works as closed loop System stops sampling when losing robust asymptotic stability, defines next sampling period corresponding to minimum optimal robustness performance γ T_s' it is the optional sampling cycle, and it is designated as T_γ。

Preferably, sampling period T is set up described in step 2_sThe process of relation between network delay τ includes following step Suddenly：

Step 2.1 determines queuing model；

Wherein, P₀For the probability containing 0 packet in queue；

P₁For the probability containing 1 packet in queue；

P_n-1For the probability containing n-1 packet in queue；

P_nFor the probability containing n packet in queue；

P_n+1For the probability containing n+1 packet in queue；

λ is the speed into queue；

μ is the speed of dequeue；

Step 2.2 state probability sum in queue is 1, supplements equation；

P_n>=0 n=0,1,2 ... m (7)

Step 2.3 can obtain the probability of stability by formula (6) and (7)：

P_n=(λ/μ)ⁿ(1-λ/μ),λ≤μ (8)

If entering the speed of queueThe speed of dequeueThen formula (8) can transform to：

P_n=(T_send/T_s)ⁿ(1-T_send/T_s),T_s≥T_send (9)

Thus sampling period T is set up_sWith the relation between network delay τ：

Preferably, the Dynamic Output Feedback robust controller described in step 3 is：

The beneficial effects of the present invention are：

1st, for the characteristics of network delay, using queuing model, establishing sampling period and net in network control system Relation between network delay；

2nd, the optional sampling cycle is determined, it is ensured that the global optimum of system, i.e., under the optimal sampling period, system reaches To optimal robustness characteristic；

3rd, Dynamic Output Feedback H_∞The design of robust controller, is easy to practical engineering application, it is ensured that system is disturbed in the external world In the case of dynamic interference, still with good stability, dynamic and robustness.

Brief description of the drawings

Fig. 1 is the flow chart of control method of the present invention.

Fig. 2 is the structure chart of network control system.

Fig. 3 is the graph of a relation in control method of the present invention between sampling period and network delay.

Fig. 4 is to adopt the graph of a relation in control method of the present invention between sample cycle and robust performance.

Embodiment

Clear, complete description is carried out to technical scheme below in conjunction with accompanying drawing.Obviously described implementation Example is only a part for the embodiment of the present invention, and based on embodiments of the invention, those skilled in the art is not making creation Property work on the premise of the other embodiments that obtain, belong to the protection domain of this patent.

The embodiment provides a kind of sampling period optimal control method of network control system, including to quilt Control object measurement output variable and the sampling by tune output variable.

According to the flow chart of the control method shown in Fig. 1, embodiment step of the present invention is as follows：

Step 1, the measurement output variable to controlled device, sampled by tune output variable, and build discrete system mould Type；

Wherein, state variable when x (k) is controlled device current time k；

X (k+1) is the state variable at controlled device next controlling cycle (k+1) moment；

Control input amount when u (k) is controlled device current time k；

Control input amount when u (k-1) is previous controlling cycle (k-1)；

Measurement output variable when y (k) is current time k；

Z (k) be current time k when by tune output variable；

External disturbance when w (k) is current time k；

For controlled device continuous time state variable x (t) coefficient, t is continuous time Variable, T_s=10ms is the sampling period；

Controlled for controlled device continuous time Input quantity u (t) coefficient, τ is network delay；

For the coefficient 1 of external disturbance；

C₁=[1 0] are state variable x (k) coefficient matrix 1；

C₂=[1 1] are state variable x (k) coefficient 2；

D₁=[0 0] are the coefficient 2 of external disturbance；

D₂=[- 1-2] are the coefficient 3 of external disturbance.

Step 2, set up sampling period T_sWith the relation between network delay τ；

Wherein, T_send=10ms is transmission cycle, and T_s≥T_send, m=10 is the length of queue, and n is contained number in queue According to the number of bag.

Specifically, the derivation of (2) formula, i.e., described to set up sampling period T_sThe mistake of relation between network delay τ Journey comprises the following steps：

1) queuing model is determined；

Wherein, P₀For the probability containing 0 packet in queue；

P₁For the probability containing 1 packet in queue；

P_n-1For the probability containing n-1 packet in queue；

P_nFor the probability containing n packet in queue；

P_n+1For the probability containing n+1 packet in queue；

λ is the speed into queue；

μ is the speed of dequeue；

2) state probability sum is 1 in queue, supplements equation；

P_n>=0 n=0,1,2 ... m (7)

3) probability of stability can be obtained by formula (6) and (7)：

P_n=(λ/μ)ⁿ(1-λ/μ),λ≤μ (8)

If entering the speed of queueThe speed of dequeueThen formula (8) can transform to：

P_n=(T_send/T_s)ⁿ(1-T_send/T_s),T_s≥T_send (9)

Thus sampling period T is set up_sWith the relation between network delay τ：

Step 3, design Dynamic Output Feedback robust controller, and build closed-loop model；

The expression formula of the closed-loop model is as follows：

Wherein, x (k-1) is the state variable at controlled device previous controlling cycle (k-1) moment；

x_c(k) state variable when for controller current time k；

x_c(k+1) it is the state variable at controller next controlling cycle (k+1) moment；

x_c(k-1) it is the state variable at controller previous controlling cycle (k-1) moment；

W (k-1) is the external disturbance at previous controlling cycle (k-1) moment；

For controller state variable x_c(k) coefficient 1；

For measurement output y (k) coefficient 1；

For controller state variable x_c(k) coefficient 2；

For measurement output y (k) coefficient 2.

During closed-loop model is built, the Dynamic Output Feedback robust controller of design is：

Whether whether the closed-loop model obtained by step 4, detecting step 3 meets robust asymptotic stability requirement, i.e., full Sufficient inequality (4), if meeting inequality (4), closed-loop model meets robust asymptotic stability requirement, into step 5, otherwise Return to step 1；

Wherein, W₅₁=A_d+B₁₀D_cC₂,W₅₁ ^TFor W₅₁Transposition；

W₅₂=B₁₀C_c,W₅₂ ^TFor W₅₂Transposition；

W₅₃=B₁₁D_cC₂,W₅₃ ^TFor W₅₃Transposition；

W₅₄=B₁₁C_c,W₅₄ ^TFor W₅₄Transposition；

W₆₁=B_CC₂,W₆₁ ^TFor W₆₁Transposition；

For A_cTransposition；

For the coefficient 1, P of liapunov function^-1For the inverse of P；

For liapunov function matrix 2, S^-1For the inverse of S；

For liapunov function matrix 3；

For liapunov function matrix 4.

Step 5, set up sampling period T_sWith the relation between optimal robustness performance γ；

Wherein, W₇₅=B₁₀D_cC₂+B₂,W₇₅ ^TFor W₇₅Transposition；For B_cTransposition；γ=3.1531 are optimal robustness Energy；ε=5.1217 × 10^-7For scalar；For unit matrix.

Step 6, setting sampling step length h=1ms, and set a sampling period T_s'=T_s+ h, repeat step 1,2,3,4,5, Stop sampling when closed-loop system loses robust asymptotic stability, define next sampling corresponding to minimum optimal robustness performance γ Cycle T_s' it is the optional sampling cycle, and it is designated as T_γ。

By above-mentioned steps, next sampling period T can be obtained_s' relation between network delay τ is as shown in figure 3, next sampling Cycle T_s' relation between optimal robustness performance γ is as shown in Figure 4.Then minimum optimal robustness performance γ=1.1822 institutes are right The next sampling period T answered_s' it is optional sampling cycle, i.e. T_γ=24ms.

Now, Dynamic Output Feedback optimal robust controller is：

Wherein, A_c=-0.0069, B_c=-0.0055,

Claims (3)

1. a kind of network control system sampling period optimal control method, including output variable is measured to controlled device and adjusted The sampling of output variable, it is characterised in that key step is as follows：

Step 1, the measurement output variable to controlled device, sampled by tune output variable, and build discrete system model；

$\left\{\begin{array}{c}x(k+1)=A_{d}x\left(k\right)+B_{10}u\left(k\right)+B_{11}u(k-1)+B_{2}w\left(k\right)\\ z\left(k\right)=C_{1}x\left(k\right)+D_{1}w\left(k\right)\\ y\left(k\right)=C_{2}x\left(k\right)+D_{2}w\left(k\right)\end{array}\right.---\left(1\right)$

Wherein,

State variable when x (k) is controlled device current time k；

X (k+1) is the state variable at controlled device next controlling cycle (k+1) moment；

Control input amount when u (k) is controlled device current time k；

Control input amount when u (k-1) is previous controlling cycle (k-1)；

Measurement output variable when y (k) is current time k；

Z (k) be current time k when by tune output variable；

External disturbance when w (k) is current time k；

A is controlled device continuous time state variable x (t) coefficient and is a constant, and t is consecutive hours anaplasia Amount, T_sFor the sampling period；

B₁For controlled device continuous time control input amount u's (t) Coefficient and be a constant, τ is network delay；

B₂Coefficient 1 for external disturbance and be a constant；

C₁Coefficient 1 for state variable x (k) and be a constant；

C₂Coefficient 2 for state variable x (k) and be a constant；

D₁Coefficient 2 for external disturbance and be a constant；

D₂Coefficient 3 for external disturbance and be a constant；

Step 2, set up sampling period T_sWith the relation between network delay τ；

$\mathrm{\τ}=\underset{n=1}{\overset{m}{\Σ}}{(T_{send}/T_s)}^n(1-T_{send}/T_s)\·n\·T_{send}+T_{send}---\left(2\right)$

Wherein, T_sendTo send cycle, and T_s≥T_send, m is the length of queue, and n is the number of contained packet in queue；

Step 3, design Dynamic Output Feedback robust controller, and build closed-loop model；

The expression formula of the closed-loop model is as follows：

$\left\{\begin{array}{c}\left[\begin{array}{c}x(k+1)\\ x_c(k+1)\end{array}\right]=\left[\begin{array}{cc}{A}_d+B_{10}{D}_c{C}_2& B_{10}{C}_c\\ B_c{C}_2& A_c\end{array}\right]\left[\begin{array}{c}x\left(k\right)\\ x_c\left(k\right)\end{array}\right]=\left[\begin{array}{cc}{B}_{11}{D}_c{C}_2& B_{11}{C}_c\\ 0& 0\end{array}\right]\left[\begin{array}{c}x(k-1)\\ x_c(k-1)\end{array}\right]\\ +\left[\begin{array}{cc}{B}_{10}{D}_c{C}_2+B_2& B_{11}{D}_c{C}_2\\ B_c{D}_2& 0\end{array}\right]\left[\begin{array}{c}w\left(k\right)\\ w(k-1)\end{array}\right]\\ z\left(k\right)=C_{1}x\left(k\right)+D_{1}w\left(k\right)\end{array}\right.---\left(3\right)$

Wherein,

X (k-1) is the state variable at controlled device previous controlling cycle (k-1) moment；

x_c(k) state variable when for controller current time k；

x_c(k+1) it is the state variable at controller next controlling cycle (k+1) moment；

x_c(k-1) it is the state variable at controller previous controlling cycle (k-1) moment；

W (k-1) is the external disturbance at previous controlling cycle (k-1) moment；

A_cFor controller state variable x_c(k) coefficient 1 and be a constant；

B_cTo measure output y (k) coefficient 1 and being a constant；

C_cFor controller state variable x_c(k) coefficient 2 and be a constant；

D_cTo measure output y (k) coefficient 2 and being a constant；

Whether whether the closed-loop model obtained by step 4, detecting step 3 meets robust asymptotic stability requirement, i.e., meet not Equation (4), if meeting inequality (4), closed-loop model meets robust asymptotic stability requirement, into step 5, otherwise returns Step 1；

$\left[\begin{array}{cccccc}R-P& 0& 0& 0& {W_{51}}^T& {W_{61}}^T\\ 0& Z-S& 0& 0& {W_{52}}^T& {A_c}^T\\ 0& 0& -R& 0& {W_{53}}^T& 0\\ 0& 0& 0& -Z& {W_{54}}^T& 0\\ W_{51}& W_{52}& W_{53}& W_{54}& -P^{-1}& 0\\ W_{61}& A_c& 0& 0& 0& -S^{-1}\end{array}\right]<0---\left(4\right)$

Wherein,

W₅₁=A_d+B₁₀D_cC₂,W₅₁ ^TFor W₅₁Transposition；

W₅₂=B₁₀C_c,W₅₂ ^TFor W₅₂Transposition；

W₅₃=B₁₁D_cC₂,W₅₃ ^TFor W₅₃Transposition；

W₅₄=B₁₁C_c,W₅₄ ^TFor W₅₄Transposition；

W₆₁=B_CC₂,W₆₁ ^TFor W₆₁Transposition；

For A_cTransposition；

P is the coefficient 1 of liapunov function and is a constant, P^-1For the inverse of P；

S is the coefficient 2 of liapunov function and is a constant, S^-1For the inverse of S；

R is the coefficient 3 of liapunov function and is a constant；

Z is the coefficient 4 of liapunov function and is a constant；

Step 5, set up sampling period T_sWith the relation between optimal robustness performance γ；

$\left[\begin{array}{ccccccccccc}R-P& 0& 0& 0& 0& 0& {W_{51}}^T& 0& C_1^T& C_2^T& 0\\ 0& Z-S& 0& 0& 0& 0& {W_{52}}^T& {A_c}^T& 0& 0& 0\\ 0& 0& -R& 0& 0& 0& {W_{53}}^T& 0& 0& 0& 0\\ 0& 0& 0& -Z& 0& 0& {W_{54}}^T& 0& 0& 0& 0\\ 0& 0& 0& 0& -{\mathrm{\&gamma;}}^{2}I& 0& {W_{75}}^T& 0& D_1^T& 0& 0\\ 0& 0& 0& 0& 0& 0& {W_{53}}^T& 0& 0& 0& 0\\ W_{51}& W_{52}& W_{53}& W_{54}& W_{75}& W_{53}& -P^{-1}& 0& 0& 0& 0\\ 0& A_c& 0& 0& 0& 0& 0& -S^{-1}& 0& 0& {\mathrm{\&epsiv;B}}_C\\ C_1& 0& 0& 0& D_1& 0& 0& 0& -I& 0& 0\\ C_2& 0& 0& 0& 0& 0& 0& 0& 0& -\mathrm{\&epsiv;}I& 0\\ 0& 0& 0& 0& 0& 0& 0& {\mathrm{\&epsiv;B}}_C^T& 0& 0& -\mathrm{\&epsiv;}I\end{array}\right]<0---\left(5\right)$

Wherein, W₇₅=B₁₀D_cC₂+B₂,W₇₅ ^TFor W₇₅Transposition；For B_cTransposition；γ is optimal robustness performance；ε is scalar；For unit matrix；

Step 6, setting sampling step length h, and set a sampling period T_s'=T_s+ h, repeat step 1,2,3,4,5, works as closed-loop system Stop sampling when losing robust asymptotic stability, define next sampling period T corresponding to minimum optimal robustness performance γ_s' be The optional sampling cycle, and it is designated as T_γ。

2. a kind of network control system sampling period optimal control method according to claim 1, it is characterised in that step Sampling period T is set up described in rapid 2_sThe process of relation between network delay τ comprises the following steps：

Step 2.1 determines queuing model；

$\left\{\begin{array}{cc}-{\mathrm{\&lambda;P}}_0+{\mathrm{\&mu;P}}_1=0,& n=0\\ {\mathrm{\&lambda;P}}_{n-1}-(\mathrm{\&lambda;}+\mathrm{\&mu;})P_n+{\mathrm{\&mu;P}}_{n+1}=0,& m\&GreaterEqual;n\&GreaterEqual;1\end{array}\right.---\left(6\right)$

Wherein, P₀For the probability containing 0 packet in queue；

P₁For the probability containing 1 packet in queue；

P_n-1For the probability containing n-1 packet in queue；

P_nFor the probability containing n packet in queue；

P_n+1For the probability containing n+1 packet in queue；

λ is the speed into queue；

μ is the speed of dequeue；

Step 2.2 state probability sum in queue is 1, supplements equation；

$\underset{n=0}{\overset{m}{\&Sigma;}}{P}_n=1,P_n\&GreaterEqual;0,n=0,1,2,...m---\left(7\right)$

Step 2.3 can obtain the probability of stability by formula (6) and (7)：

P_n=(λ/μ)ⁿ(1-λ/μ),λ≤μ (8)

If entering the speed of queueThe speed of dequeueThen formula (8) can transform to：

P_n=(T_send/T_s)ⁿ(1-T_send/T_s),T_s≥T_send (9)

Thus sampling period T is set up_sWith the relation between network delay τ：

$\mathrm{\&tau;}=\underset{n=1}{\overset{m}{\&Sigma;}}{(T_{send}/T_s)}^n(1-T_{send}/T_s)\&CenterDot;n\&CenterDot;T_{send}+T_{send}---\left(2\right).$

3. a kind of network control system sampling period optimal control method according to claim 1, it is characterised in that step Dynamic Output Feedback robust controller described in rapid 3 is：

$\left\{\begin{array}{c}{x}_c(k+1)=A_c{x}_c\left(k\right)+B_{c}y\left(k\right)\\ u\left(k\right)=C_c{x}_c\left(k\right)+D_{c}y\left(k\right)\end{array}\right.---\left(10\right).$