This post is based on Chapter 4 of “Feedback Control of Dynamic Systems (third edition)” by Franklin, Power, and Emami-Naeini. I we have an eigenvector c. We need to generate a fourth eigenvector, d. The only eigenvalue that needs another eigenvector is Together we then get a state-space realization with matrices A, B and C determined by the strictly proper part, and matrix D determined by the constant. To figure out which one works, we need to plug both vectors into the generating equation: If a generates the correct vector d, we will order our eigenvectors as: but if b generates the correct vector, we can order it as: If a matrix has a complete set of distinct eigenvectors, the transition matrix T can be defined as the matrix of those eigenvectors, and the resultant transformed matrix will be a diagonal matrix. ⋅

= λ For


$$v_i(t)=Ri(t)+L\frac{\text{d}i(t)}{\text{d}t}+v_c(t)$$, $$\Rightarrow \frac{\text{d}i(t)}{\text{d}t}=-\frac{Ri(t)}{L}-\frac{v_c(t)}{L}+\frac{v_i(t)}{L}$$. can be continuous (e.g. A The output equation gives the relationship between the system’s current state and input to its output. For instance, in a simple mechanical mass-spring-damper system, the two state variables could be the position and velocity of the mass. The trick here is to separate the transfer function into two parts: a strictly proper part and a constant. The next code block is used to simulate the system response to non-zero initial conditions. 1 q r = K , {\displaystyle \lambda =2} If we have a non-singular n × n matrix P, we can define a transformed vector "x bar" as: We can transform the entire state-space equation set as follows: We call the matrix P the equivalence transformation between the two sets of equations. Z This is due to the The terms "Eigenvalues" and "Eigenvect… t The state space formulation of a set of differential equations is easier to solve with a digital computer. solving the output equation for A state mode for this system is not unique but depends on the choice of a set of state variables x. 1 Figure 1(b) shows a free body diagram of the rotor.

"item": On the other hand, the damping torque can be expressed as follows: where is a damping coefficient. These facts give rise to several results: For those reasons, the system eigenvalues and the condition number of the system matrix are highly important variables to consider when analyzing and designing a system.
The code lines 13-16 are used to define the system matrices. λ y = x, which yields C = I, the Identity matrix. D is the direct transmission matrix of size (m×r) X(t) is the state vector of size (n×1) Y(t) is the output vector of size (m×1) U(t) is the input vector of size (r×1) (Block diagram of the linear, continuous time control system represented in state space) � =� ï� �+ð� � � �=ñ� �+ò� � 0

. and substituting in the state equation

Many applications rely on the Kalman Filter to produce estimates of the current unknown state variables using their previous observations.[11][12]. In this general formulation, all matrices are allowed to be time-variant (i.e. The system step response is important for understanding the dynamics of the system.

p


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