Mathematics

Questions tagged [linear-control]

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

568 questions
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Stability of linear time-varying systems

Consider the following linear time-varying (LTV) system $$\dot{x} = A(t)x$$ If $A(t)$ satisfies $$\mbox{eig} \left( A(t)+A(t)^{T} \right) < 0$$ then is it sufficient to conclude that the time-varying system is stable? I am looking for references…
jjgarrison
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10
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2 answers

Singular values of a matrix written in controllable canonical form

Let the following equation represent a stable (marginally) dynamical system in discrete time domain \begin{equation} \mathbf{x}_{k+1} = \mathbf{A}\mathbf{x}_k + \mathbf{B}\mathbf{u}_k \end{equation} where $\mathbf{A}\in\mathbb{R}^{m\times m}$,…
user146290
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9
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Maximize function of eigenvalues related to a quadratic Lyapunov function

I would like to solve the following problem $$ \begin{align} \max_Q\ & \frac{\lambda_\text{min}(Q)}{2\,\lambda_\text{max}(P)} \sqrt{\frac{\lambda_\text{min}(P)}{\lambda_\text{max}(P)}} \\ s.t.\ & P\,A + A^\top P = -Q \\ & Q = Q^\top \succ 0…
Kwin van der Veen
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9
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0 answers

Two definitions of uniformly observable, are they equivalent?

When I do my research, I found that there are two definitions of uniformly observable, I can't help thinking are they equivalent? These two definitions are listed as follows. For a linear stochastic system with the description $$x_{k+1} =…
wayne
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1 answer

Is the controllability Gramian always positive definite?

I am trying to understand the balanced truncation algorithm and have some trouble distinguishing between controllability matrix and controllability Gramian. If my understanding is correct, a linear time invariant system $\dot x(t) = Ax(t) + Bu$ is…
milez
  • 235
7
votes
0 answers

All equivalent inverse LQR problems

Inspired by this question I wondered if it is possible to fully parameterize the inverse optimal control problem. So given a stabilizing state feedback policy $$ u(t) = -K\,x(t), \tag{1} $$ for a linear time invariant state space system $$…
Kwin van der Veen
  • 12,870
7
votes
2 answers

Wrong stability results when using Padé approximation

The Padé approximation of the exponential function, $F(s) = e^{-\tau s}$ , is used often in control theory. I wonder whether its use can lead to erroneous results regarding the stability properties of the closed-loop system. Let's say we have a…
Nicolas Bert Johnson
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6
votes
4 answers

Riccati & Lyapunov equations

We know that Lyapunov equation: $A^TP + PA + Q = 0$ Algebraic Riccati equation: $A^TP + PA + Q + PBR^{-1}B^TP= 0$ It seems that the difference between the two lies in $B = 0$ (zero input) in Lyapunov Eq and both are infinite horizon in the case…
sleeve chen
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6
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1 answer

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 definite matrix $B\in\mathbb{R}^{n\times n}$…
graham
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6
votes
1 answer

Relation between **poles** of $\boldsymbol{G}(s)$ and **eigenvalues** of system matrix $A$.

In page 602 of [1], it states: It is also possible to relate zeros to the properties of a coprime factorization of a given transfer function matrix. In particular, if we write \begin{equation} \boldsymbol{G}(s) =…
winston
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6
votes
1 answer

Proof general state space similarity transformation to controllable canonical form

Given a state space model of the form, $$ \begin{align} \dot{x} &= A\,x + B\,u \\ y &= C\,x + D\,u \end{align} \tag{1} $$ however I think that this would also apply to a discrete time model. Assuming that this state space model is controllable, I…
Kwin van der Veen
  • 12,870
6
votes
3 answers

Why can't an improper transfer function be realized?

A major result in control system theory is that a transfer function, $$G\left( s \right) = \frac{{Y\left( s \right)}}{{U\left( s \right)}}$$ has a state space realization if and only if the degree of $Y(s)$ is less than or equal to the degree of…
ITA
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5
votes
1 answer

Kalman Filter with correlated observation noise and identity observation matrix

My Kalman Filter model is very simple, I have a identity transition matrix and identity observation matrix. State transition: $x_t = x_{t-1} + v_t$ where $v_t \sim N(0,Q)$ . The states kind of follow a random walk (there is still correlation between…
Geroge Klein
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5
votes
1 answer

State-space with second derivative input variable

I have a second order ODE with derivative terms $$ \alpha_1\frac{d^2i}{dt^2}+\alpha_2\frac{di}{dt}+\alpha_3i=-\beta_1V_r-\beta_2\frac{dV_r}{dt}-\beta_3\frac{d^2V_r}{dt^2} $$ I want to transform this system into state-space, but I'm not sure how to…
TurboChad
  • 71
5
votes
3 answers

Proving that similarity transformation of state-space preserves the Euclidean norm

I am aware that the state space realization of a dynamical system is not unique. So if we have a dynamical system: $\dot{x} = Ax + Bu$ $y = Cx$ Then we can write it as $\begin{bmatrix} \dot{x}\\y \end{bmatrix} = G \begin{bmatrix}x\\u\end{bmatrix} $…
user782248
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