Fundamental Concepts

State Space Model

Given a system, we have various methods to define its properties, including State Space Model,
Transfer function, Odinary differential equations and etc. Different description methods can be interchanged.

Example

Given a CT(Continuous-time) or DT(Discrete-time) system,

CT:x˙(t)=Ax(t)+Bu(t)y(t)=Cx(t)+Du(t)DT:x(t+1)=Ax(t)+Bu(t)y(t)=Cx(t)+Du(t)\begin{split} \mathrm{CT}: \quad & \dot{x}(t) = Ax(t) + Bu(t) \\ & y(t) = Cx(t) + Du(t) \\ \mathrm{DT}: \quad & x(t+1) = Ax(t) + Bu(t) \\ & y(t) = Cx(t) + Du(t) \end{split}

Their transfer functions are,

G(s)=C(sI−A)−1B+D,LaplaceG(z)=C(zI−A)−1B+D,z−transfrom\begin{split} G(s) & = C(sI-A)^{-1}B+D, \quad \mathrm{Laplace} \\ G(z) & = C(zI-A)^{-1}B+D, \quad \mathrm{z-transfrom} \end{split}

Notice that transfer fucntion descirbes properties of the sytem in frequency domain.

Details about state space model could be seen at this link: State Space Model

Response of the system

Now we connect the system with their response in time domain. Given a sytem in state space model with their inital condition x0x_0 and input u(t)u(t). Their general response satisfy the following equations

y(t)=CeAtx0+∫0tCeA(t−τ)Bu(τ)dτ+Du(τ)(CT)y(t)=CeAtx0+Σi=0t−1CAt−1−iBu(i)+Du(t)(DT)\begin{split} y(t) & = Ce^{At}x_0 + \int_0^tCe^{A(t-\tau)}Bu(\tau)d\tau+Du(\tau) \quad \mathbb{(CT)}\\ y(t) & = Ce^{At}x_0 + \Sigma_{i=0}^{t-1}CA^{t-1-i}Bu(i)+Du(t) \quad \mathbb{(DT)} \end{split}

Impulse reponse

Impulse reponse generally refers to the output of the system when the input is a unit pulse function. In other words, the initial condition of the sytem x0=0x_0=0 and input u(t)=δ(t)u(t)=\delta(t).

y(t)=CeAtB+Dδ(t)∣y(t)=CAt−1By(t) = Ce^{At}B+D\delta(t) \quad | \quad y(t)=CA^{t-1}B

A simple method to compute eAte^{At}

Given a state space model, if we would like to know its response in time domain. It’s needed to calculate eAte^{At}. A simple method to calculate is computing (sI−A)−1(sI-A)^{-1} or z(zI−A)−1z(zI-A)^{-1} then use inverse Laplace or Z-transform.

Canonical Form

Given a transfer function G(s)G(s), we can use Canonical Form to derive its state space model.

G(s)=b0sn+b1sn−1+...+bn−1s+bnsn+a1sn−1+...+an−1s+an=b0+c1sn−1+...+cn−1s+cnsn+a1sn−1+...+an−1s+an\begin{split} G(s) & = \frac{b_0s^n+b_1s^{n-1}+...+b_{n-1}s+b_n}{s^n+a_1s^{n-1}+...+a_{n-1}s+a_n} \\ & = b_0 + \frac{c_1s^{n-1}+...+c_{n-1}s+c_n}{s^n+a_1s^{n-1}+...+a_{n-1}s+a_n} \end{split}

with ci=bi−b0ai,i=1,2,...,nc_i=b_i-b_0a_i, i= 1, 2,..., n

Then we can get the controllability Canonical Form

A=[010…0001…0⋮⋮⋮⋮⋮000…1−an−an−1−an−2…−a1]B=[00⋮01]C=[cncn−1…c2c1]D=[b0]\begin{split} A & = \begin{bmatrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \dots & 1 \\ -a_n & -a_{n-1} & -a_{n-2} & \dots & -a_1 \end{bmatrix} \\ B & = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \end{bmatrix} \\ C & = \begin{bmatrix} c_n & c_{n-1} & \dots & c_2 & c_1 \end{bmatrix} \\ D & = [b_0] \end{split}