# How to design the right input to obtain a desired output for a linear system?

If I have a state-space model, so that matrices $$A$$, $$B$$, $$C$$ and $$D$$ are known, how can I design the right input $$u$$, so that $$y$$ is a desired signal, say, a sine wave with constant amplitude?

\begin{aligned} \dot{x} &= A x + B u \\ y &= C x + D u \end{aligned}

• Please add some information about A-D. Since your equation system contains derivatives character of the matrices influences possible ‚design‘ approaches Jan 17, 2018 at 19:00
• A, B, C and D are real values matrices. A is 18x18, B is 18x2, C is 2x18, D is 2x2. I do not care about x and its derivative. It is a 2x2 state space model but the cross-coupling terms are zero, so the input u1 will not influence y2 and so on. For example, the B matrix is: non-zero on the first nine elements of the first column; non-zero on the last nine elements of the second column. Jan 18, 2018 at 7:31
• @Alessandro But is $A$ Hurwitz, the state space model minimal (so controllable and observable) and is $D$ full rank? Jan 19, 2018 at 13:52
• D is a 2x2 and has rank 2. A, indeed, is 18x18 but has rank 16, so the system is observable neither controllable. Jan 19, 2018 at 15:39

• That is a very crappy control approach. If the model used was $\frac{1}{s+1}$, the controller would be $s+1$. There are two problems: 1) It is not a proper system. 2) If the actual plant was $\frac{1}{s+1.1}$, the plant will controller will actually be $\frac{s+1}{s+1.1}$ instead of 1. So the tracking will have errors. On paper everything should be fine, and you should not be seeing the problem you are describing. Jan 19, 2018 at 14:10