Question

i dont understand this problem. please show how to solve all parts using MATLAB. thank you.

State-Space Representation and Analysis csys canon(sys,type) compute a canonical state-space realization type 'companion': controllable canonical form type modal: modal canonical form poles of a system controllability matrix observability matrix eig(A) ctrb(A,B) obsv(A,C) -7 L-12 0 EX A 2C-ioD0 uestions () Define the system in the state-space form (2) Determine the stability of the system (3) Determine the controllability and the observability of the system. 4) Convert the system to controllable and modal canonical forms.

Answer 1

The MATLAB code and the necessary comments are provide:

A=[-7 1;-12 0]; %creates matrix A
B=[1;2]; %creates matrix B
C=[1 0]; %creates matrix C
D=0; %creates matrix D
sys=ss(A,B,C,D); %creates state space model of the system
poles_sys=eig(sys); %obtain the poles of the system
Cn=ctrb(A,B); %if Cn has the rank of 2 (rank of A) system is controllable
Co=obsv(A,C); %if Co has the rank of 2 (rank of A) system is observable
csys=canon(sys,'companion'); %controllable canonical form
osys=canon(sys,'modal'); %modal canonical form

The poles of the system given by poles_sys are -4 and -3. Since they lie on the Left hand side of s-plane system is stable

Rank of Cn given by the function rank(Cn) is 2 and so system is controllable. Rank of Co given by the function rank(Co) is 2 and so system is observable.