State Equation Fundamentals

1Introduction

This lecture represents a system's internal state as a vector and uses a state equation to analyze its evolution, output, and stability under a given input.

2State and State-Space Representation

The stateState is sufficient internal information whose value at a given time, together with the subsequent input, uniquely determines the system's future evolution. A state choice need not be unique, and a representation may contain redundant variables. For finite-dimensional LTI systems, any two minimal realizations of the same input-output relation are related by a similarity transformation.

A continuous-time LTI state-space representationState-space representation is

For state $\mathbf{x}\in R^n$\mathbf x\in\mathbb R^n, input $\mathbf{u}\in R^m$\mathbf u\in\mathbb R^m, and output $\mathbf{y}\in R^p$\mathbf y\in\mathbb R^p,

$A$A represents coupling among states, $B$B the action of inputs, $C$C the map from state to output, and $D$D the direct term from input to output.

3Construction from a Second-Order Equation

Consider the system

with mass $m>0$m>0, damping coefficient $c\text{[PARSE ERROR: Undefined("Command("ge")")]}0$c\ge 0, and spring constant $k\text{[PARSE ERROR: Undefined("Command("ge")")]}0$k\ge 0. Define $x_1=q$x_1=q and $x_2=\dot{q}$x_2=\dot q. Then

Position and velocity have different physical dimensions, so the entries of $A$A do not all have the same units. Each row of the product $A\mathbf{x}$A\mathbf x, however, has the same physical dimension as the corresponding state derivative. This dimensional consistency provides a model check.

4State Transition

When $A$A and $B$B are constant matrices, the solution for initial state $\mathbf{x}\left(0\right)=x_0$\mathbf x(0)=\mathbf x_0 is

The state-transition matrixState-transition matrix $e^{At}$e^{At} describes zero-input state evolution, while the integral describes the contribution of the input history.

5Eigenvalues and Internal Stability

The origin of the zero-input system $\dot{\mathbf{x}}=A\mathbf{x}$\dot{\mathbf x}=A\mathbf x is asymptotically stableAsymptotically stable if solutions starting sufficiently near the origin remain near it and satisfy $\mathbf{x}\left(t\right)\to 0$\mathbf x(t)\to0 as $t\to \mathrm{\infty }$t\to\infty.

For a finite-dimensional continuous-time LTI system, all eigenvalues of $A$A lying in the open left half-plane, $\mathrm{Re}\lambda <0$\operatorname{Re}\lambda<0, is equivalent to global exponential stability of the origin. An eigenvalue with positive real part makes the origin unstable. If an eigenvalue lies on the imaginary axis, solutions may remain bounded or may diverge according to the Jordan structure, but the origin is not asymptotically stable.

6Relationship to the Transfer Function

With zero initial state, take Laplace transforms and consider $s$s for which $sI-A$sI-A is invertible. Then

and hence

Here, $G\left(s\right)\in C^{p\times m}$G(s)\in\mathbb C^{p\times m} is the transfer-function matrix. A mode that cannot be excited through the input is an uncontrollable modeUncontrollable mode; a mode that does not appear in the output is an unobservable modeUnobservable mode. If a state-space representation contains either type, an eigenvalue of $A$A may not appear as a pole of $G\left(s\right)$G(s). A transfer function alone therefore need not reveal every internal stability property.

7Key Points

• State is internal information that, together with the future input, determines the future response.
• The dimensions of $A,B,C,D$A,B,C,D follow from the dimensions of state, input, and output.
• $e^{At}$e^{At} describes zero-input evolution, and the eigenvalues of $A$A determine internal stability.
• $G\left(s\right)=C(sI-A{)}^{-1}B+D$G(s)=C(sI-A)^{-1}B+D is a zero-initial-state input-output relation and need not expose every internal mode.