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Introduction to matlab, 6.057 introduction to matlab, homework 1.

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Matlab Programming for Numerical Computation

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

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Course layout.

The course will be covered in eight modules. Various aspects of MATLAB programming for numerical computation will be covered in these modules, with each module dedicated to on equivalent numerical topic. Each module will be covered in one week, with 2–2.5 hours lectures per week. There will be self-study problems at the end of several of these lectures. Assignments will also be posted periodically.

Module 1: Introduction to MATLAB Programming

Module 2: Approximations and Errors

Module 3: Numerical Differentiation and Integration

Module 4: Linear Equations

Module 5: Nonlinear Equations

Module 6: Regression and Interpolation

Module 7: Ordinary Differential Equations (ODE) – Part 1

Module 8: Ordinary Differential Equations (ODE) – Practical aspects

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Instructor bio.

Prof. Niket Kaisare

Prof. Niket Kaisare is a Professor of Chemical Engineering in IIT-Madras. He works in the area of modeling, design and control for energy applications. He has over ten years of research/teaching experience in academia, and three-year experience in Industrial R&D. He uses computational software, including MATLAB, FORTRAN, Aspen and FLUENT extensively in his research and teaching. Faculty web-page:   http://www.che.iitm.ac.in/~nkaisare/

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Course Name: MATLAB Programming for Numerical Computation

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Course abstract

MATLAB is a popular language for numerical computation. This course introduces students to MATLAB programming, and demonstrate it?s use for scientific computations. The basis of computational techniques are expounded through various coding examples and problems, and practical ways to use MATLAB will be discussed. The objective of this course is to introduce undergraduate students to computational methods using MATLAB. At the end of this course, a student would: Learn basics of MATLAB programming Get introduced to numerical methods for engineering problems Will be able to use MATLAB to solve computational problems

Course Instructor

Prof. Niket Kaisare

Dr. Niket Kaisare is an Associate Professor of Chemical Engineering in IIT-Madras. He works in the area of modeling, design and control for energy applications. He has over 5 years of research/teaching experience in academia, and three-year experience in Industrial R&D. He uses computational software, including MATLAB, FORTRAN, Aspen and FLUENT extensively in his research and teaching.

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 course duration : jan-mar 2021,   view course,  syllabus,  enrollment : 18-nov-2020 to 25-jan-2021,  exam registration : 15-jan-2021 to 12-feb-2021,  exam date : 21-mar-2021,   course statistics will be published shortly, certificate eligible, certified category count, successfully completed, participation.

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• Assignment Score = Average of best 6 out of 8 assignments. Final Score(Score on Certificate)= 75% of Exam Score + 25% of Assignment Score. Note: We have taken best assignment score from both Jan 2020 and Jan2021 course

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Total enrollment: 4708, registration statistics, total registration : 380, assignment statistics, score distribution graph - legend, assignment score: distribution of average scores garnered by students per assignment., exam score : distribution of the final exam score of students., final score : distribution of the combined score of assignments and final exam, based on the score logic., feedback videos.

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असाइनमेंट meaning in hindi

[सं-पु.] - किसी विशेष घटना या समाचार से संबंधित समाचार के संकलन हेतु संवाददाता को सौंपी गई जिम्मेदारी।

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Pole placement design

Description

Pole placement is a method of calculating the optimum gain matrix used to assign closed-loop poles to specified locations, thereby ensuring system stability. Closed-loop pole locations have a direct impact on time response characteristics such as rise time, settling time, and transient oscillations. For more information, see Pole Placement .

From the figure, consider a linear dynamic system in state-space form

x ˙ = A x + B u

y = C x + D u

For a given vector p of desired self-conjugate closed-loop pole locations, place computes a gain matrix K such that the state feedback u = – Kx places the poles at the locations p . In other words, the eigenvalues of A – BK will match the entries of p (up to the ordering).

K = place( A , B , p ) places the desired closed-loop poles p by computing a state-feedback gain matrix K . All the inputs of the plant are assumed to be control inputs. place also works for multi-input systems and is based on the algorithm from [1] . This algorithm uses the extra degrees of freedom to find a solution that minimizes the sensitivity of the closed-loop poles to perturbations in A or B .

[ K , prec ] = place( A , B , p ) also returns prec , an accuracy estimate of how closely the eigenvalues of A – BK match the specified locations p ( prec measures the number of accurate decimal digits in the actual closed-loop poles). A warning is issued if some nonzero closed-loop pole is more than 10% off from the desired location.

collapse all

Pole Placement Design for Second-Order System

For this example, consider a simple second-order system with the following state-space matrices:

A = [ - 1 - 2 1 0 ] B = [ 2 0 ] C = [ 0 1 ] D = 0 Spate-space matrices

Input the matrices and create the state-space system.

Compute the open-loop poles and check the step response of the open-loop system.

Notice that the resultant system is underdamped. Hence, choose real poles in the left half of the complex-plane to remove oscillations.

Find the gain matrix K using pole placement and check the closed-loop poles of syscl .

Now, compare the step response of the closed-loop system.

Hence, the closed-loop system obtained using pole placement is stable with good steady-state response.

Note that choosing poles that are further away from the imaginary axis achieves faster response time but lowers the steady-state gain of the system. For instance, consider using the poles [-2,-3] for the above system.

Pole Placement Precision

For this example, consider the pole locations [-2e-13,-3e-4,-3e-3] . Compute the precision of the actual poles.

A precision value of 2 is obtained indicating that the actual pole locations are precise up to 2 decimal places.

Pole Placement Using Complex Poles

For this example, consider the following transfer function with complex-conjugate poles at - 2 ± 2 i :

s y s t f ( s ) = 8 s 2 + 4 s + 8 Transfer function of the system

Input the transfer function model. Then, convert it to state-space form since place uses the A and B matrices as input arguments.

Next, compute the gain matrix K using the complex-conjugate poles.

The values of the gain matrix are real since the poles are self-conjugate. The values of K would be complex if p did not contain self-conjugate poles.

Now, verify the step response of the closed-loop system.

Pole Placement Observer Design

For this example, consider the following SISO state-space model:

A = [ - 1 - 0 . 7 5 1 0 ] B = [ 1 0 ] C = [ 1 1 ] D = 0 SISO State-Space Model

Create the SISO state-space model defined by the following state-space matrices:

Now, provide a pulse to the plant and simulate it using lsim . Plot the output.

For this example, assume that all the state variables cannot be measured and only the output is measured. Hence, design an observer with this measurement. Use place to compute the estimator gain by transposing the A matrix and substituting C' for matrix B . For this instance, select the desired pole locations at -2 and -3 .

Use the estimator gain to substitute the state matrices using the principle of duality/separation and create the estimated state-space model.

Simulate the time response of the system using the same pulse input.

Compare the response of the actual system and the estimated system.

Input Arguments

A — state matrix nx -by- nx matrix.

State matrix, specified as an Nx -by- Nx matrix where, Nx is the number of states.

B — Input-to-state matrix Nx -by- Nu matrix

Input-to-state matrix, specified as an Nx -by- Nu matrix where, Nx is the number of states and Nu is the number of inputs.

p — Closed-loop pole locations vector

Closed-loop pole locations, specified as a vector of length Nx where, Nx is the number of states. In other words, the length of p must match the row size of A . Closed-loop pole locations have a direct impact on time response characteristics such as rise time, settling time, and transient oscillations. For an example on selecting poles, see Pole Placement Design for Second-Order System .

place returns an error if some poles in p have multiplicity greater than rank(B) .

In high-order problems, some choices of pole locations result in very large gains. The sensitivity problems attached with large gains suggest caution in the use of pole placement techniques. See [2] for results from numerical testing.

Output Arguments

K — optimum gain ny -by- nx matrix.

Optimum gain or full-state feedback gain, returned as an Ny -by- Nx matrix where, Nx is the number of states and Ny is the number of outputs. place computes a gain matrix K such that the state feedback u = – Kx places the closed-loop poles at the locations p .

When the matrices A and B are real, K is

real when p is self-conjugate.

complex when the pole locations are not complex-conjugates.

prec — Accuracy estimate of the assigned poles scalar

Accuracy estimate of the assigned poles, returned as a scalar. prec measures the number of accurate decimal digits in the actual closed-loop poles in contrast to the pole locations specified in p .

You can use place for estimator gain selection by transposing the A matrix and substituting C' for matrix B as follows, as shown in Pole Placement Observer Design . You can use the resultant estimator gain for state estimator workflows using estim .

[1] Kautsky, J., N.K. Nichols, and P. Van Dooren, "Robust Pole Assignment in Linear State Feedback," International Journal of Control, 41 (1985), pp. 1129-1155.

[2] Laub, A.J. and M. Wette, Algorithms and Software for Pole Assignment and Observers , UCRL-15646 Rev. 1, EE Dept., Univ. of Calif., Santa Barbara, CA, Sept. 1984.

Version History

Introduced before R2006a

lqr | rlocus | estim

• Pole Placement

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