M 2 K
Author: m | 2025-04-24
m = 4, k = 3, m^2 9k^2 k = 100, Square root = 10 m = 6, k = 5, m^2 9k^2 - k = 256, Square root = 16 m = 6, k = 7, m^2 9k^2 k = 484, Square root = 22 m = 8, k = 9 K/m - (r^2)/(4m^2) = (K/m)1 - (K/m)(m/K)(r^2)/(4m^2) [here I rewrote each term with (K/m) as a factor] [here I canceled an m] = (K/m)[1 - (r^2)/(4mK)] [here I multiplied the
$ sum_{k=1}^m k(k-1){m choose k} = m(m-1) 2^{m-2}$
+ K i K f ) ] G 3 = L m m ema + m el L m + J m L m η ema k ema 2 R G 2 = η ema k ema 2 R ( J m R m + J m K i K f + B fm L m ) + B fr L m + ( m ema + m el ) ( R m + K i K f ) G 1 = η ema k ema 2 R ( C T C E + B fm R m + K i K f B fm ) + B fr ( R m + K i K f ) + S t L m G 0 = S t ( R m + K i K f ) By dividing both the numerator and the denominator on the right side of Equation (11) by the coefficient of u, we obtain: F ema = u − ( F 3 s 3 + F 2 s 2 + F 1 s ) x a ( s 2 m el + S t ) R k ema η ema K i K v C T [ ( s 2 m el + S t ) R k ema η ema K i K v C T ] ( G 3 s 3 + G 2 s 2 + G 1 s + G 0 ) (12) It can be seen from Figure 13 that to suppress the disturbance force, the feedforward correction element should be used to counteract the part following u in the numerator of Equation (12): G T = F 3 s 3 + F 2 s 2 + F 1 s ( s 2 m el + S t ) R k ema η ema K i K v C T (13) The simulation model of the DEMA force servo system introduced with a PID controller and a feedforward correction element is shown in Figure 14 [48,49,50].As shown in Figure 14, we added several modules in the model to simulate the sampling and data processing of the processor, making the simulation model more realistic. In order to make the simulation model easier to understand and operate, we use PMSM and planetary-roller screw pairs as super components, and integrate feedforward and its sampling and data processing into the super components.The disturbance of
Is k m ? (1) m = 4(k 1) (2) m = k 2 : Data Sufficiency (DS)
Disturbance forces on the EMA, thereby badly affecting the accuracy and dynamic performance of the force servo system. Therefore, the force servo system should suppress disturbance forces to the greatest extent possible. In other words, the EMA force servo system should have sufficient anti-disturbance capability, rapidity, and loading accuracy.In this study, drawing on the principle of structure invariance, a relatively complex high-order transfer function GT associated with various parameters such as speed and acceleration was added as feedforward compensation to xa, so as to suppress the impact of external position disturbances on the system’s motion performance. The system model block diagram with the feedforward correction element added is shown in Figure 13.Since the stiffness Ks of the force sensor is normally taken as 5 × 107 N/m, much larger than the connection stiffness St between the LEMA and the load connection point, the effect of the stiffness Ks of the force sensor on the system’s stability and dynamic characteristics can be neglected when calculating the controller parameters [45]. It can be approximated as a rigid connection, and thus the open-loop transfer function of the force servo system can be simplified as follows: F ema = R k ema η ema K i K v C T u − s F ab F ema + S t x a s 2 m el + S t s L m + R m + K i K f = ( s 2 m el + S t ) R k ema η ema K i K v C T u − ( F 3 s 3 + F 2 s 2 + F 1 s ) x a G 3 s 3 + G 2 s 2 + G 1 s + G 0 (11) where: F 3 = S t ( L m m ema + R J m L m η ema k ema 2 ) F 2 = S t [ η ema k ema 2 R ( J m R m + J m K i K f + B fm L m ) + B fr L m + m ema ( R m + K i K f ) ] F 1 = S t [ η ema k ema 2 R ( C T C E + B fm R m + K i K f B fm ) + B fr ( R mSolve for m K=(mv^2)/2
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T ⋯ x 2 + ( m − 1 ) t ⋮ ⋮ ⋯ ⋮ x M x M + t ⋯ x N ] . (1) At this time, the correlation integral is Formula (2), where θ ( x ) = { 0 x 0 1 x ≥ 0 . According to the statistical conclusion of BDS, the range of M and r k can be obtained when N > 3000 ; m ∈ { 2 , 3 , 4 , 5 } , and r k = k × 0.5 σ which is a real number representing a given range of distances. σ is the standard deviation of time series, and k ∈ { 1 , 2 , 3 , 4 } . Correlation integral indicates the probability that the distance between any two points in the phase space is less than r k . C ( m , N , r k , t ) = 2 M ( M − 1 ) ∑ 1 ≤ i j ≤ M θ ( r k − ‖ X i − X j ‖ ) . (2) We define the test statistic S and Δ S , and use the block averaging strategy, as shown in Formula (3). { S ( m , N , r k , t ) = 1 t ∑ i = 1 M C i ( m , N t , r k , t ) − C i m ( m , N t , r k , t ) Δ S ( m , N , t ) = max [ S ( m , N , r k , t ) ] − min [ S ( m , N , r k , t ) ] . (3) Formula (4) is the average of S and Δ S . Rounding the t value of the first zero of S ¯ or the first minimum of Δ S is the optimal delay topt. { S ¯ ( t ) = 1 4 × 4 ∑ m = 2 5 ∑ k = 1 4 S ( m , N , r k , t ) Δ S ¯ ( t ) = 1 4 ∑ m = 2 5 Δ S ( m , N , t ) . (4) Formula (5) is the test statistic. The global minimum of S c o r ( t ) is the optimal embedded window t ω . S c o r ( t ) = Δ S ¯ ( t ) + | S ¯ ( t ) | . (5) Then: t ω = ( m o p t − 1 ) t o p t . (6) Therefore, the optimal delay topt determined by Equation (4) and the optimal embedded window t ω determined by Equation (5) can be substituted into Formula (6) and rounded to obtain the optimal embedding dimension mopt. 2.2. Deep Neural NetworkA neural network generally consists ofwhat is the value of k (2k m) when k=-2 and m=7 - brainly.com
Problem 681For a square matrix $M$, its matrix exponential is defined by\[e^M = \sum_{i=0}^\infty \frac{M^k}{k!}.\]Suppose that $M$ is a diagonal matrix\[ M = \begin{bmatrix} m_{1 1} & 0 & 0 & \cdots & 0 \\ 0 & m_{2 2} & 0 & \cdots & 0 \\ 0 & 0 & m_{3 3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & m_{n n} \end{bmatrix}.\]Find the matrix exponential $e^M$. Add to solve laterSponsored Links Solution.First, we find $M^k$ for each integer $k \geq 0$. The first couple powers can be calculated directly,\[M^0 = \begin{bmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{bmatrix} , \quad M = \begin{bmatrix} m_{1 1} & 0 & 0 & \cdots & 0 \\ 0 & m_{2 2} & 0 & \cdots & 0 \\ 0 & 0 & m_{3 3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & m_{n n} \end{bmatrix},\]\[M^2 = \begin{bmatrix} m^2_{1 1} & 0 & 0 & \cdots & 0 \\ 0 & m^2_{2 2} & 0 & \cdots & 0 \\ 0 & 0 & m^2_{3 3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & m^2_{n n} \end{bmatrix} , \quad M^3 = \begin{bmatrix} m^3_{1 1} & 0 & 0 & \cdots & 0 \\ 0 & m^3_{2 2} & 0 & \cdots & 0 \\ 0 & 0 & m^3_{3 3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & m^3_{n n} \end{bmatrix}.\]The general pattern can now be seen:\[M^k = \begin{bmatrix} m^k_{1 1} & 0 & 0 & \cdots & 0 \\ 0 & m^k_{2 2} & 0 & \cdots & 0 \\ 0 & 0 & m^k_{3 3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & m^k_{n n} \end{bmatrix}.\]Now, we can calculate the infinite series $e^M$:\begin{align*}e^M &= \sum_{k=0}^{\infty} \frac{ M^k }{k!} \\&= \sum_{k=0}^\infty \frac{1}{k!} \begin{bmatrix} m^k_{1, 1} & 0 & 0 & \cdots & 0 \\ 0 & m^k_{2, 2} & 0 & \cdots & 0 \\ 0 & 0 & m^k_{3, 3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & m^k_{n, n} \end{bmatrix} \\&= \begin{bmatrix} \sum_{k=0}^\infty \frac{ m^k_{1 1} }{k!} & 0 & 0 & \cdots & 0 \\ 0 & \sum_{k=0}^\infty \frac{ m^k_{2 2} }{k!} & 0 & \cdots & 0 \\ 0 & 0 & \sum_{k=0}^\infty \frac{ m^k_{3If sum _{ k=1 }^{ m }{ ({ k }^{ 2 }1)(k!)=2025(2025!) }, then the
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Amlogic S905 (ARM Cortex-A53), 1536 MHz, (28 nm). 4 cores. 2 GB DDR3-1824 (13-13-13) (32-bit). ODROID-C2 board.L1 Data cache = 32 KB, 64 B/line, 4-WAY, PIPT.L1 Instruction cache = 32 KB, 64 B/line, 2-WAY, VIPT. L2 Cache = 512 KB, 64 B/line, 16-WAY, shared by all cores.Branch Target Address Cache: 256-entry.Branch predictor: 3072-entry pattern history prediction table.Return stack : 8-entry. L1 Data Cache Latency = 3 cycles for simple access via pointer L1 Data Cache Latency = 3 cycles for access with complex address calculation (size_t n, *p; n = p[n]). L2 Cache Latency = 15 cycles RAM Latency = 15 cycles + 128 ns2 MB pages (64-bit Linux) Micro Data TLB L1 (4 KB pages): 10 items. ?-way. Miss penalty = 2 cycles. Parallel miss: 3 cycles per L1 data cache read access. TLB L2 (2 MB pages): 512 items. 4-way. Miss penalty = 20 ? cycles. Size Latency Increase Description 32 K 3 64 K 10 6 + 12 (L2) + 2 (Micro TLB miss) 128 K 14 4 256 K 16 2 512 K 17 1 1 M 17 + 70 ns + 70 ns + 128 ns (RAM) 2 M 17 + 100 ns + 30 ns 4 M 17 + 114 ns + 14 ns 8 M 17 + 121 ns + 7 ns 16 M 17 + 124 ns + 3 ns 32 M 17 + 126 ns + 2 ns 64 M 17 + 127 ns + 1 ns 128 M 17 + 128 ns + 1 ns 256 M 17 + 128 ns 512 M 17 + 128 ns 1024 M 17 + 128 ns 2048 M + 20 ? (L2 TLB miss) 4 KB pages (64-bit Linux) Micro Data TLB L1 (4 KB pages): 10 items. ?-way. Miss penalty = 2 cycles. Parallel miss: 3 cycles per L1 data cache read access. TLB L2: 512 items. 4-way. Miss penalty = 11-22 cycles (translation entry from L1/L2 cache). Parallel miss: 22 cycles per L2 cache read access. Size Latency Increase Description 32 K 3 64 K 10 6 + 12 (L2) + 2 (Micro TLB miss) 128 K 14 4 256 K 16 2 512 K 17 + 4 ns 1 + 4 ns 1 M 17 + 70 ns + 66 ns + 128 ns (RAM) 2 M 17 + 100 ns + 30 ns 4 M 25 + 114 ns 8 + 14 ns + 22 (L2 TLB miss) 8 M 31 + 121 ns 6 + 7 ns 16 M 35 + 124 ns 4 + 3 ns 32 M 37 + 126 ns 2 + 2 ns 64 M 39 + 127 ns 2 + 1 ns 128. m = 4, k = 3, m^2 9k^2 k = 100, Square root = 10 m = 6, k = 5, m^2 9k^2 - k = 256, Square root = 16 m = 6, k = 7, m^2 9k^2 k = 484, Square root = 22 m = 8, k = 9Solve for n m/(np^2)=k
Computing the three-dimensional FFT. However, the size of the original array is modified to contain one or two additional rows, which are needed to store the values identically equal to zero. The values of the arguments used with the real three-dimensional FFT routines depend upon whether an in-place or out-of place transform is performed, and whether the results are stored in a full or partial result matrix, as shown in TABLE 19. TABLE 19 Relationship Between Values of Arguments for Real Three-Dimensional FFT Routines Full Result Array Partial Result Array In-Place Transform B unused B unused LDB unused LDB unused LDA must be even LDA must be even LDA 2*M LDA M+2 if M is evenLDA M+1 if M is odd A(1:2*M, 1:N) A(1:M+2, 1:N) if M is evenA(1:M+1, 1:N) if M is odd Out-of-Place Transform A unchanged A unchanged LDA M LDA M LDB 2*M LDB M/2+1 if M is evenLDB (M-1)/2+1 if M is odd B(1:2*M, 1:N, 1:K) B(1:M+2, 1:N, 1:K) if M is evenB(1:M+1, 1:N, 1:K) if M is odd When computing the real 3D FFT of an input sequence of M rows, N columns, and K planes, the computed Fourier coefficients will be stored in a result matrix with 2*M rows, N columns for each value of K when using the Full storage option. When using the Partial storage option, the Fourier coefficients will be stored in a result matrix with M+2 rows and N columns for each value of K when M is even, orComments
+ K i K f ) ] G 3 = L m m ema + m el L m + J m L m η ema k ema 2 R G 2 = η ema k ema 2 R ( J m R m + J m K i K f + B fm L m ) + B fr L m + ( m ema + m el ) ( R m + K i K f ) G 1 = η ema k ema 2 R ( C T C E + B fm R m + K i K f B fm ) + B fr ( R m + K i K f ) + S t L m G 0 = S t ( R m + K i K f ) By dividing both the numerator and the denominator on the right side of Equation (11) by the coefficient of u, we obtain: F ema = u − ( F 3 s 3 + F 2 s 2 + F 1 s ) x a ( s 2 m el + S t ) R k ema η ema K i K v C T [ ( s 2 m el + S t ) R k ema η ema K i K v C T ] ( G 3 s 3 + G 2 s 2 + G 1 s + G 0 ) (12) It can be seen from Figure 13 that to suppress the disturbance force, the feedforward correction element should be used to counteract the part following u in the numerator of Equation (12): G T = F 3 s 3 + F 2 s 2 + F 1 s ( s 2 m el + S t ) R k ema η ema K i K v C T (13) The simulation model of the DEMA force servo system introduced with a PID controller and a feedforward correction element is shown in Figure 14 [48,49,50].As shown in Figure 14, we added several modules in the model to simulate the sampling and data processing of the processor, making the simulation model more realistic. In order to make the simulation model easier to understand and operate, we use PMSM and planetary-roller screw pairs as super components, and integrate feedforward and its sampling and data processing into the super components.The disturbance of
2025-03-31Disturbance forces on the EMA, thereby badly affecting the accuracy and dynamic performance of the force servo system. Therefore, the force servo system should suppress disturbance forces to the greatest extent possible. In other words, the EMA force servo system should have sufficient anti-disturbance capability, rapidity, and loading accuracy.In this study, drawing on the principle of structure invariance, a relatively complex high-order transfer function GT associated with various parameters such as speed and acceleration was added as feedforward compensation to xa, so as to suppress the impact of external position disturbances on the system’s motion performance. The system model block diagram with the feedforward correction element added is shown in Figure 13.Since the stiffness Ks of the force sensor is normally taken as 5 × 107 N/m, much larger than the connection stiffness St between the LEMA and the load connection point, the effect of the stiffness Ks of the force sensor on the system’s stability and dynamic characteristics can be neglected when calculating the controller parameters [45]. It can be approximated as a rigid connection, and thus the open-loop transfer function of the force servo system can be simplified as follows: F ema = R k ema η ema K i K v C T u − s F ab F ema + S t x a s 2 m el + S t s L m + R m + K i K f = ( s 2 m el + S t ) R k ema η ema K i K v C T u − ( F 3 s 3 + F 2 s 2 + F 1 s ) x a G 3 s 3 + G 2 s 2 + G 1 s + G 0 (11) where: F 3 = S t ( L m m ema + R J m L m η ema k ema 2 ) F 2 = S t [ η ema k ema 2 R ( J m R m + J m K i K f + B fm L m ) + B fr L m + m ema ( R m + K i K f ) ] F 1 = S t [ η ema k ema 2 R ( C T C E + B fm R m + K i K f B fm ) + B fr ( R m
2025-03-31T ⋯ x 2 + ( m − 1 ) t ⋮ ⋮ ⋯ ⋮ x M x M + t ⋯ x N ] . (1) At this time, the correlation integral is Formula (2), where θ ( x ) = { 0 x 0 1 x ≥ 0 . According to the statistical conclusion of BDS, the range of M and r k can be obtained when N > 3000 ; m ∈ { 2 , 3 , 4 , 5 } , and r k = k × 0.5 σ which is a real number representing a given range of distances. σ is the standard deviation of time series, and k ∈ { 1 , 2 , 3 , 4 } . Correlation integral indicates the probability that the distance between any two points in the phase space is less than r k . C ( m , N , r k , t ) = 2 M ( M − 1 ) ∑ 1 ≤ i j ≤ M θ ( r k − ‖ X i − X j ‖ ) . (2) We define the test statistic S and Δ S , and use the block averaging strategy, as shown in Formula (3). { S ( m , N , r k , t ) = 1 t ∑ i = 1 M C i ( m , N t , r k , t ) − C i m ( m , N t , r k , t ) Δ S ( m , N , t ) = max [ S ( m , N , r k , t ) ] − min [ S ( m , N , r k , t ) ] . (3) Formula (4) is the average of S and Δ S . Rounding the t value of the first zero of S ¯ or the first minimum of Δ S is the optimal delay topt. { S ¯ ( t ) = 1 4 × 4 ∑ m = 2 5 ∑ k = 1 4 S ( m , N , r k , t ) Δ S ¯ ( t ) = 1 4 ∑ m = 2 5 Δ S ( m , N , t ) . (4) Formula (5) is the test statistic. The global minimum of S c o r ( t ) is the optimal embedded window t ω . S c o r ( t ) = Δ S ¯ ( t ) + | S ¯ ( t ) | . (5) Then: t ω = ( m o p t − 1 ) t o p t . (6) Therefore, the optimal delay topt determined by Equation (4) and the optimal embedded window t ω determined by Equation (5) can be substituted into Formula (6) and rounded to obtain the optimal embedding dimension mopt. 2.2. Deep Neural NetworkA neural network generally consists of
2025-04-02Problem 681For a square matrix $M$, its matrix exponential is defined by\[e^M = \sum_{i=0}^\infty \frac{M^k}{k!}.\]Suppose that $M$ is a diagonal matrix\[ M = \begin{bmatrix} m_{1 1} & 0 & 0 & \cdots & 0 \\ 0 & m_{2 2} & 0 & \cdots & 0 \\ 0 & 0 & m_{3 3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & m_{n n} \end{bmatrix}.\]Find the matrix exponential $e^M$. Add to solve laterSponsored Links Solution.First, we find $M^k$ for each integer $k \geq 0$. The first couple powers can be calculated directly,\[M^0 = \begin{bmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{bmatrix} , \quad M = \begin{bmatrix} m_{1 1} & 0 & 0 & \cdots & 0 \\ 0 & m_{2 2} & 0 & \cdots & 0 \\ 0 & 0 & m_{3 3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & m_{n n} \end{bmatrix},\]\[M^2 = \begin{bmatrix} m^2_{1 1} & 0 & 0 & \cdots & 0 \\ 0 & m^2_{2 2} & 0 & \cdots & 0 \\ 0 & 0 & m^2_{3 3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & m^2_{n n} \end{bmatrix} , \quad M^3 = \begin{bmatrix} m^3_{1 1} & 0 & 0 & \cdots & 0 \\ 0 & m^3_{2 2} & 0 & \cdots & 0 \\ 0 & 0 & m^3_{3 3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & m^3_{n n} \end{bmatrix}.\]The general pattern can now be seen:\[M^k = \begin{bmatrix} m^k_{1 1} & 0 & 0 & \cdots & 0 \\ 0 & m^k_{2 2} & 0 & \cdots & 0 \\ 0 & 0 & m^k_{3 3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & m^k_{n n} \end{bmatrix}.\]Now, we can calculate the infinite series $e^M$:\begin{align*}e^M &= \sum_{k=0}^{\infty} \frac{ M^k }{k!} \\&= \sum_{k=0}^\infty \frac{1}{k!} \begin{bmatrix} m^k_{1, 1} & 0 & 0 & \cdots & 0 \\ 0 & m^k_{2, 2} & 0 & \cdots & 0 \\ 0 & 0 & m^k_{3, 3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & m^k_{n, n} \end{bmatrix} \\&= \begin{bmatrix} \sum_{k=0}^\infty \frac{ m^k_{1 1} }{k!} & 0 & 0 & \cdots & 0 \\ 0 & \sum_{k=0}^\infty \frac{ m^k_{2 2} }{k!} & 0 & \cdots & 0 \\ 0 & 0 & \sum_{k=0}^\infty \frac{ m^k_{3
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