# 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 – rul30 Jan 17 '18 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. – Alessandro Jan 18 '18 at 7:31
• @Alessandro But is $A$ Hurwitz, the state space model minimal (so controllable and observable) and is $D$ full rank? – fibonatic Jan 19 '18 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. – Alessandro Jan 19 '18 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. – Suba Thomas Jan 19 '18 at 14:10