"Question on designing a state observer for discrete time system I came through this problem while studying for an exam in control systems: Consider the following discrete time system x vec (k+1)=Ax vec (k)+bu vec (k),y vec (k)=cx vec (k) where b=(0,1)^T,c=(1,0),A [201−g] for some g in R Find a feedback regulation (if there is any) of the form u(okay)=−Kxhat(ok) where xhat(k) is the country estimation vector that is produced via a linear complete-order state observer such that the nation of the system and the estimation blunders e(ok)=xvec (k)−xhat(okay) go to zero after a few finite time. layout the kingdo observer and the block diagram.

zakownikbj 2022-09-26 Answered
Question on designing a state observer for discrete time system
I came through this problem while studying for an exam in control systems:
Consider the following discrete time system
x ( k + 1 ) = A x ( k ) + b u ( k ) , y ( k ) = c x ( k )
where b = ( 0 , 1 ) T , c = ( 1 , 0 ) , A = [ 2 1 0 g ] for some g R
Find a feedback regulation (if there is any) of the form u ( k ) = K x ^ ( k ) where x ^ ( k ) is the country estimation vector that is produced via a linear complete-order state observer such that the nation of the system and the estimation blunders e ( k ) = x ( k ) x ^ ( k ) go to zero after a few finite time. layout the kingdom observer and the block diagram.
My method
it is clean that the eigenvalues of the machine are λ 1 = 2 , λ 2 = g (consequently it is not BIBO solid) and that the pair (A,b) is controllable for every fee of g, as nicely a the pair (A,c) is observable for all values of g. consequently we will shift the eigenvalues with the aid of deciding on a benefit matrix okay such that our device is strong, i.e. it has its eigenvalues inside the unit circle | z | = 1.
The state observer equation is
[ x ( k + 1 ) e ( k + 1 ) ] T = [ A b K B k O A L C ] [ x ( k ) e ( k ) ] T
With characteristic equation
χ ( z ) = | z I A + b K | | z I A + L C | = χ K ( z ) χ L ( z )
Also consider
K = [ k 1 k 2 k 3 k 4 ]
and let a = k 1 + k 3 , β = k 2 + k 4
Then χ K ( z ) = ( z 2 ) ( z + g + β ) + a.
So we can select some eigenvalues inside the unit circle and determine a , β in terms of g. Choosing e.g. λ 1 , 2 = ± 1 / 2 we get a = 3 g + 33 / 8 , β = 9 / 4 g , g R
Questions
I want to ask the following:
Is my approach correct? Should I select the eigenvalues myself since I am asked to design the observer or should I just solve the characteristic equation and impose | λ 1 , 2 | < 1?
Should I determine L matrix as well since the error must also vanish? (because it is not asked)
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Kaya Garza
Answered 2022-09-27 Author has 8 answers
Your approach is right. In short, find a dynamic compensator for the observer and change coordinates so that you have the state and estimation error. That matrix will be upper triangular. Your result looks correct.
In order to design the observer and compensator gains, remember that the eigenvalues of an upper triangular matrix are on the diagonal. For a block-UT matrix, the eigenvalues are the eigenvalues of the diagonal blocks. Thus you only need your A b K and A L C to be Schur for the system to satisfy your requirements.
You can do this using pole placement or guess and check.
Since the matrices are 2 × 2 either of these will be simple to do.
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-01-31
The centers for Disease Control reported the percentage of people 18 years of age and older who smoke (CDC website, December 14, 2014). Suppose that a study designed to collect new data on smokers and Questions Navigation Menu preliminary estimate of the proportion who smoke of .26.
a) How large a sample should be taken to estimate the proportion of smokers in the population with a margin of error of .02?(to the nearest whole number) Use 95% confidence.
b) Assume that the study uses your sample size recommendation in part (a) and finds 520 smokers. What is the point estimate of the proportion of smokers in the population (to 4 decimals)?
c) What is the 95% confidence interval for the proportion of smokers in the population?(to 4 decimals)?
asked 2022-09-02
Book Recommendation for Second Course in Linear Algebra
I only took a non-rigorous linear algebra course (It was designed for non-math students). I finished most of Hungerford's algebra. Now I have two choices to study more advanced linear algebra: Hoffman's linear algebra, Finite dimensional vector spaces by Halmos. I already looked at the first chapter of both books.
I think Halmos goes at a faster pace which is something I like. Would I miss anything if I start learning the book of Halmos instead of Hoffman's ?
asked 2022-09-03
Applications of logic in sciences
I understand math as the study and description of the behavior of mathematical structures, and as you know this math structures could include rings, fields, metric spaces, propositions, categories, numbers, sets, operators, differential equations..., a big part of this structures was born under the need of the description of a problem. For example the study and solution of the problem of the Brachistochrone curve gives to us the calculus of variations, or the study of the behavior of the heat and waves was the main column of the development of the Fourier series expansion, and as you know this is useful in physics, electrical, mechanical, and in general engineering.
So other structures such as differential equations, tensors, matrices are useful for physics, chemistry, economics, engineering, and even abstract ones such as linear spaces, groups, rings, operators, Banach spaces, Hausdorff spaces are useful in physics.
But in general logic, understanding it as the classification of truth parametrized but several specifications using several structures such as languages, binary operators, models, this to proof under what conditions a given expression id true, so it has several applications in number theory, algebra, topology, but this ones are mathematical fields, so i want to know if besides computer science foundations, type theory in CS, programming languages fundamentals, design and analysis of algorithms, digital logic, computer architecture, (that by itself is a huge approach of logic in life), are there any applications of logic in physics, economics, engineering, biology..
asked 2021-03-02
In the 1970s a study was conducted in Philadelphia in which 500 cases were randomly assigned to treatments for the common cold: 250 subjects received the medication and 250 received a placebo. A total of 383 patients improved within 24 hours. Of those who received the medication 241 improved within 24 hours and of those who received the placebo 142 improved within 24 hours. A test of significance was conducted on the following hypotheses.
Ho: The rates for the two treatments are equal.
Ha: The treatment of medication has a higher improvement rate.
This test resulted in a p-value of 0.0761.
a.) Interpret what this p-value measures in the context of this study.
b.) Based on this p-value and study design, what conclusion should be drawn in the context of this study? Use a significance level of α=0.05.
c.) Based on your conclusion in part (b), which type of error, Type I or Type II, could have been made? What is one potential consequence of this error?
asked 2022-09-24
where do combinatorics play a role in computer science?
I am graduated from computer science. I need some advice in Combinatorics material related to master fields of computer science. I already know that discrete mathematics have a key role in some concepts which I need a little more elaboration. In which fields could I trace combinatorics concepts and theorems and techniques?
asked 2022-09-03
Optimization Problem : Dumpster
I am trying to help my friend. This is his problem related to constructing a dumpster, so it can minimize construction cost :
For this project we locate a trash dumpster in order to study its shape and construction. We then attempt to determine the dimensions of a container of similar design that minimize construction cost."
(Already located, measured, and described a dumpster found).
"While maintaining the general shape and method of construction, determine the dimensions such a container of the same volume should have in order to minimize the cost of construction. Use the following assumptions in your analysis:
The sides, back, and front are to be made from 12-gauge (0.1046 inch thick) steel sheets, which cost $0.70 per square foot (including any required cuts or bends).
The base is to be made from a 10-gauge (0.1345 inch thick) steel sheet, which costs $0.90 per square foot.
Lids cost approximately $50.00 each, regardless of dimensions.
Welding costs approximately $0.18 per square foot for material and labor combined.
Give justification of any further assumptions or simplifications made of the details of construction.
Describe how any of your assumptions or simplifications may affect the actual result.
If you were hired as a consultant on this investigation, what would your conclusion be? Would you recommend altering the design of the dumpster? If so, describe the savings that would result."
asked 2022-09-04
Find out the angular speed in terms of time.
Here is the equation that describes the motion of a planet under the gravitational field generated by a fixed star: u = e l cos θ + 1 l , where u is the reciprocal of the radial distance between the planet and the star, e is the eccentricity of the orbit, l is the semi latus rectum, and let h denote the angular momentum per unit mass, θ is the angular coordinate. e,h,l turn out to be independent from one another, and they are independent from t and θ. At time t=0, we let the radial speed vanish, and we also let the angular coordinate vanish. To find the relationship between time and angular speed ω, we assume that u is a smooth fuction of t, and differentiate u w.r.t. t, and use ω = h u 2 to find out an expression for ω. To do this we can differentiate u w.r.t. θ first then multiply it by ω, which equals to h u 2 .
Then differentiate the first derivate of u w.r.t θ first, then multiply the result by hu2 and so on. Since the whole process involves the differentiation w.r.t. θ only, we can assume that e=0.5,l=h=1. We set e=0.5 only because we wish to study bounded orbits so that we can apply Kepler's law and verify our result. However, the whole process is time consuming since the formula for the derivatives of u becomes complicated very quickly, even if we assume explicit values for e,l,h. The only effective way, therefore, is to design an algorithm for this process. But I do not have any knowlege about computer science, if anyone knows how to design algorithms or know about some other ways to find out ω in terms of t, please share.

New questions

i'm seeking out thoughts for a 15-hour mathematical enrichment course in a chinese language high faculty. What (pretty) simple concern would you advocate as a subject for any such course?
historical past/issues:
My students are generally pretty good at math, but many of them have no longer been uncovered to rigorous or summary mathematical reasoning. an amazing topic would be one that could not be impossibly hard for students who have by no means written or study proofs in English.
i have taught this magnificence three times earlier than. (a part of the purpose that i'm posting that is that i have used up all my thoughts!) the primary semester I taught an introductory range theory elegance (which meandered its way toward a proof of quadratic reciprocity, though I think this become in the end too advanced/abstract for some of the students). the second one semester I taught fundamental graph idea and packages (with a focal point on planarity and coloring). The 1/3 semester I taught a class at the Rubik's dice.
the students' math backgrounds are pretty numerous: a number of them take part in contest math competitions, and so are familiar with IMO-fashion techniques, however many aren't. a number of them may additionally realize some calculus, however I cannot assume it. all of them are superb at what in the united states is on occasion termed "pre-calculus": trigonometry, conic sections, systems of linear equations (though, shockingly, no matrices), and the like. They realize what a binomial coefficient is.
So, any ideas? preferably, i'd like to find some thing a bit "sexy" (like the Rubik's cube) -- tries to encourage wide variety theory through cryptography seemed to fall on deaf ears, however being capable of "see" institution idea on the cube became pretty popular.
(Responses specifically welcome from folks who grew up in the percent -- any mathematical subjects you desire were protected within the excessive college curriculum?)