Questions tagged [linear-control]

Linear control theory is the sub-branch of control theory dealing with linear or linearized systems.

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120 views

Markov Decision Processes and Control Engineering?

In a Markov Decision Process, one has a Markov chain with (left) regular stochastic matrix $P$ and a collection of "actions" $a_i$ one can have act on the system after it transitions via its ...
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Balancing Transformation of Linear Time Invariant Systems

I've been experimenting with linear time invariant systems. In particular I've been playing around with the concept of balanced truncation. All the resources I've found only explain how to do balanced ...
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Proving a transfer function is proper or not

I'm trying to prove (not just show) whether the following transfer function is proper or not $$ G(s) = \frac{1}{1+se^{-s}} $$ A transfer function $G(s)$ is said to be proper if there is $\alpha \geq 0$...
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Model building a container

in this question I'm supposed to make an equation for the height of fluid in the container the container is cylinder form with base area of $A_0 =1261 m^2$. The variables are $V$ inflow stream. $W$ ...
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31 views

Do these two expressions describe the same $L^2$ norm?

In a lecture on model order reduction, the following expression is said to be equivalent to the $L^2$ norm of $h(t): \mathbb{R} \rightarrow \mathbb{R}^{r \times m}$, if $h(t) \rightarrow 0$ for $t \...
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51 views

Coercivity of integral of solution to Lyapunov differential equation

Let us consider the integral operator $T:\mathbb{R}^{n\times d}\to [0,\infty)$ such that for all $K\in \mathbb{R}^{n\times d}$, $$ T(K)=\int_0^1 \operatorname{tr}(KK^\top \Sigma_t) \,d t, $$ where $\...
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75 views

Show $\lim \limits_{t \to \infty} x(t) = 0$ exponentially for any $x(0)\in \mathbb{R}^n$ if $\lim \limits_{t \to \infty} u(t) = 0$ exponentially.

I'd like to ask a question about the controllability of a linear system. Any help would be appreciated. $$$$Given a vector $x(t)\in \mathbb{R}^n$ defined over [$0$, $\infty$), we say $x(t)$ converge ...
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99 views

Inverse dynamics control: Proof of asymptotic stability of error system

The inverse dynamics control in robotic applications yields the error system \begin{equation} \ddot{\mathbf{e}} + \mathbf{K}_1 \dot{\mathbf{e}} + \mathbf{K}_0 {\mathbf{e}} = \mathbf{0} \end{equation} ...
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24 views

Closed form solution to operator norm minimization?

Is it possible to give a closed form solution to the value or minimizer to the following problem? $$ \inf \{ \|AX + B\|_{\rm op} : \|X\|_{F} = 1\} $$ Above, $A, X, B$ are square matrices, and $\|\cdot\...
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Should a linear controller be internally stable?

Suppose we have a SISO plant with the transfer function $G(s)$, and $Y(s) = G(s)U(s)$, where $Y(s)$ and $U(s)$ are the images of the output $y(t)$ and input $u(t)$, respectively. We design a SISO ...
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An LMI transformation problem

Given matrices $Q>0,F,A$ and a number $\alpha\in(0,1)$, find some $P>0,X,\Psi$ such that $$ \begin{aligned} \Psi^T P \Psi\leq \alpha P\\ \begin{bmatrix} A^TP+PA-...
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94 views

Lyapunov equation with semidefinite right-hand side

Consider the Lyapunov equation $$A^TX+XA=-Q$$ with Hurwitz matrix $A$ and positive semi-definite matrix $Q\succeq0$. When is its solution $X$ strictly positive definite?
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60 views

Why not use global optimization algorithms like PSO to solve decentralized control problems?

I do not see many works use global optimization algorithms to solve decentralized control problems. Here the decentralized control problem means some entries of the feedback matrix are constrained to ...
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121 views

Linearization of the ODE system: Problems

I have summarized the issues covered in the topics: https://mathematica.stackexchange.com/questions/253133/linearization-of-ode-without-an-equilibrium https://mathematica.stackexchange.com/questions/...
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53 views

Spectral radius bound for a rank-1 update

Let $A \in \mathbb{R}^{n\times n}$ and let $u,v \in \mathbb{R}^{n\times 1}$. Suppose we know the following: $$ A\text{ has an eigenvalue at $1$},\quad \rho(A) = 1, \quad\text{and}\quad \rho(A + uv^\...
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52 views

Finite-Time Stability Concept

I am trying to understand the concept of finite-time stability. I found some articles that cover controller design to satisfy finite-time stability of a system but these were actually so complex. What ...
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56 views

Optimal static feedback gain matrix for stabilization

Consider a system $$ \dot{x}(t) = (A+KC)x(t) + K w(t) $$ where $w(t)\in\mathbb{R}^m$ is an unknown disturbance; $A\in\mathbb{R}^{n\times n}$ and $C\in\mathbb{R}^{m\times n}$ are known matrices with $(...
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68 views

How can I do Model Reference Adaptive Control for MIMO systems?

This is MRAC - Model Reference Adaptive Control for SISO systems. $G_m(s)$ is our reference model. It's is a first order system because they don't have overshoot. $G_m(s)$ is a desired wish how then ...
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32 views

Does a definition exist for observability for a particular number of time steps?

Consider the state space representation of the following discrete time system: $$ x_{k+1} = A x_k + B x_k \\ y_{k} = C x_k $$ The system is observable if the row rank of the observability matrix $\...
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Convergence proof of iterative linear quadratic regulator (iLQR)

Background The trajectory optimization problem can be expressed as: \begin{align} \min_{\mathbf{u}_{1}, \mathbf{u}_{1}, \ldots, \mathbf{u}_{T}} & \sum_{t=1}^{T} g(\mathbf{x}_{t}, \mathbf{u}_{t})\\ ...
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Large scaled finite-horizon discrete-time LQR

Background The standard finite-horizon discrete-time LQR is to minimize the quadratic cost below: \begin{align} \min_{\mathbf{u}_{1}, \mathbf{u}_{2}, \ldots, \mathbf{u}_{T - 1}} J & = \mathbf{x}_{...
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28 views

Upper bound on spectral radius of sum of two Metzler matrices

Given a Metzler matrix $A$ (non-diagonal elements are positive), I am trying to find an upper bound on the spectral radius of $A+A^T$ (preferably in terms of the spectral radius of $A$). In particular,...
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37 views

Selection of a representative operating point for the design of a PID controller for nonlinear systems

As described in the source, there are several ways to control a nonlinear system with a linear PID controller. If no fixed operating point is considered, advanced methods like gain-scheduling should ...
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29 views

Can someone help untangle this block diagram?

I have this big block diagram that I need to turn into a transfer function, but I don't know how. I try to solve this by this, then this. I can's see what I have done wrong, but I am wrong because the ...
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91 views

State-space representation of heat equation?

I have the heat equation : \begin{equation} \frac{\partial u\left(x,y,t\right)}{\partial t}=\alpha\nabla^2 u\left(x,y,t\right)+\beta I\left(x,y\right) \label{eq:HE} \end{equation} as $u$ to be the ...
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68 views

Differential Lyapunov equation

Given the continuous-time differential Lyapunov equation $$\mathbf{A}(t)^T\mathbf{P}(t)+\mathbf{P}(t)\mathbf{A}(t)+\dot{\mathbf{P}}(t)=-\mathbf{Q}(t)$$ where $\mathbf{A}(t)$ and $\mathbf{Q}(t)$ are ...
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319 views

Observability matrix: What is its rank?

I'm writing a controller for an inverted pendulum in which the state vector isn't completely measurable. The state vector is $\mathbf{x}\in\mathbb{R}^4$ and the output vector is $\mathbf{y}\in\mathbb{...
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79 views

Linear infinite optimization problem - when are KKT conditions sufficient for an optimum

I am trying to solve the following problem: $\max \int_0^T F(t, x(t)) dt$ s.t. $G(t,x(t)) \geq 0$ and $x(t) \in [0,1]$ for all $t\in[0,T]$ where F and G are smooth functions and affine in $x(t)$. (If ...
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74 views

Linearized input affine system

For a control problem given by the nonlinear dynamics $\dot x=f(x,u)$, we may express the state space by declaring $z=(x,u)^⊤$ with $\dot u = v$ and writing $\begin{bmatrix}\dot x \\ \dot{u}\end{...
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55 views

Given only a system transfer function, after changing it using state feedback, can we know if the resulting system is controllable? Observable?

Practice test question Given $$ G(s) = \frac{(s-2)(s-5)}{(s+1)(s-3)(s+4)} $$ Use state feedback to change it to $$ \frac{(s-5)}{(s+1)(s+4)}$$ Is the resulting system controllable? Observable? If no, ...
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132 views

Stability of a discrete-time closed-loop

I modeled a dynamic system like so: $$ \dot{y} = au, $$ i.e. as an integrator. Every $T = 0.1$ seconds, the measurement is updated and remains constant in between. This motivated me to model it as a ...
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25 views

Finding transfer function of a temperature model, complexity in algebra.

Here is the equations, I'm trying to find a transfer function relating Ti (internal temperature, output) with Ta (outside temperature, input), I know the differential equation will equate to a first ...
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32 views

linearization on Matlab for Optimal control

I am studying a tutorial about pendulum-on-a-cart control on youtube and its purpose is to stabilise the pendulum at the upper position where $\theta=\pi$. The state-vector is $$\mathbf{x}=\begin{...
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33 views

Boundedness of optimal input of scalar system

Context: Given is the scalar system $$\dot{x}=-ax+u$$ with $x(0)=x_0$ and $|u(t)|\leq M$ for a constant $a$ and a positive constant $M$ along with cost criterion $$J(u)=x(1)+\frac{1}{2}\int_{0}^{1}u^2(...
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68 views

Open Loop Transfer Function and Stability?

Given the following open loop transfer function $$G_{OL} = \frac{k}{(s+1)^2(0.5s + 1)^2}e^{-\tau s}$$ Let $k = k_{max}/2$. Let $\tau_{max}$ denote the value of the parameter $\tau$ such that the ...
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Is the similarity transform used in Kalman decomposition orthogonal?

In control theory, the Kalman decomposition is used to decompose a system, so the observable and controllable states can be distinguished between the unobservable and uncontrollable states. Consider ...
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71 views

Linear control theory and transition matrix problem

So I want to prove the following statement: Consider an $n$-dimensional time-varying system $\dot{x} = A(t)x$, where $A(t)$ is continuous. If $A^{\intercal}(t)=-A(t)$ $\forall t\in\mathbb{R}$, then $\...
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Evaluate derivative of reachability Gramian: is this correct?

So I have the reachability Gramian matrix for a linear time-invariant system: \begin{align} W(t_{0},t) = \int_{t_{0}}^{t}e^{A(t-s)}BB^{\intercal}e^{A^{\intercal}(t-s)}. \end{align} In this case I have ...
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107 views

Nyquist diagram not encircle the origin

Given the function: $\frac{s-5}{s^2-4s +5}$ The function have one zero and two poles in the RHP. So according to principle of argument (or Nyquist diagram), the Nyquist plot encircle the origin N = |Z-...
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Show the sum of a sequence of trace is convergent. $\operatorname{trace}( \sum_{t=0}^{\infty} (A')^t C' \Sigma_S C A^t S)$

I encountered this problem when doing research. Matrices are all real-valued. $\Sigma_S$ is a covariance matrix. One can assume $\Sigma_S$ is positive definite. $A$ is a square matrix with dimension $...
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How to find a unknown values of state space model matrices

I have the following state-space model $$ A= \begin{bmatrix} k & 0 & 1 \\ 1 & -3 & 2 \\ 0 & 0 & -4 \\ \end{bmatrix} B= \begin{bmatrix} 0 \\ 0\\ ...
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109 views

A problem about positive definite matrices

Given $A\in\mathbb{R}^{n\times n}$, show that all eigenvalues of A has negative real part if and only if for each positive definite matrix $C\in\mathbb{R}^{n\times n}$, there exists an unique positive ...
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169 views

Show that controllability and observability are not affected by replacing A with (A+αI )

I have been asked to show that controllability and observability are not affected by replacing $A$ with $(A+αI)$. And also to show that this is not necessarily true for stabilizability. I have thought ...
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Reachability as linear map and energy ellipsoid

EDIT: In one of Steve Brunton's video lectures which I really like, he designs this energy ellipsoid. The description about it (as I perceive it), is that: even if a system is not globally ...
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110 views

Defining Lie bracket intuitively looking at commuting flows

I'm reading "Control Theory from the Geometric Viewpoint" of Agrachev. He comments: "It is natural to suggest that a lower-order term in the Taylor expansion of $(1.12)$ at $t = s = 0$ ...
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115 views

Transfer function for amplitude modulation

What is the transfer function of the amplitude modulation element where the output is simply the input multiplied by a sinusoidal wave with a constant amplitude?
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139 views

Intuition for solution to Lyapunov equation $AU + UA^T = -C$

Background: A discrete-time equation that arises in control is apparently the matrix equation: $$ U + A^TUA = - C, $$ where $A$ and $C$ are given real square matrices and the variable is $U$. It is ...
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147 views

Find $K$ equal to the eigenvalues of $A - B K$

I've a little trouble with this old exam question. It is a multiple choice exam question! A discrete state-space is represented as $$x(t-1) = Ax(t) + Bu(t)$$ where $$A = \begin{bmatrix}-3&1&-...
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173 views

Determine controllability with big A matrix in linear system

On the linear system below: \begin{aligned} \dot{x} &= Ax + Bu \\ \end{aligned} where $$ A = \begin{bmatrix} 1 & 0 &0 &-1 \\ 0 & 1 &2 &-1 \\ 1 &...
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413 views

discrete-time to continuous-time state space [closed]

What are the formulas to convert discrete-time state space ($A_d,B_d,C_d,D_d$) to continuous-time state space ($A,B,C,D$)? i.e., Convert the state space $x_{k+1} = A_d x_{k} + B_d u_k$ $y_k = C_d x_{k}...

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