# Unexpected uncontrollable double integrator system during reference tracking (augmented state-space)

I'm trying to design a controller that will achieve reference tracking of the second state for the following system.

\begin{aligned} \dot{x} &=\underbrace{\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right]}_A x+\underbrace{\left[\begin{array}{l} 0 \\ 1 \end{array}\right]}_B u \\ y &=\underbrace{\left[\begin{array}{ll} 1 & 0 \end{array}\right]}_C x \end{aligned} where $$x=\left[\begin{array}{l} q \\ \dot{q} \end{array}\right]$$ This problem should be equivalent to tracking a ramp on the first state, hence we require a type-2 system if we are to achieve zero steady-state error. Therefore, we require a double integrator to augment the system with the integrals of the error between the reference and performance output. The error $$e$$, reference $$r$$ (step) and performance matric $$H = \left[\begin{array}{l} 0 & 1 \end{array}\right]$$ are related as $$e=r-Hx$$ If $$e=0$$, the reference is tracked with zero steady-state error. The system can be augmented by adding states $$\int e$$ and $$\int \int e$$, with dynamics $$\dot{w}=\underbrace{\left[\begin{array}{l} 0 & 0 \\ 1 & 0 \end{array}\right]}_Mw+\underbrace{\left[\begin{array}{l} 1 \\ 0 \end{array}\right]}_Ne$$ where $$w=\left[\begin{array}{l} \int e \\ \int \int e \end{array}\right]$$ The augmented system is then given by $$\left[\begin{array}{l} \dot{x} \\ \dot{w} \end{array}\right]=\underbrace{\left[\begin{array}{l} A & 0_{2 \times 2}\\ -NH & M \end{array}\right]}_{A_e}\left[\begin{array}{l} x \\ w \end{array}\right]+\underbrace{\left[\begin{array}{l} B \\ 0_{2 \times 1} \end{array}\right]}_{B_e}u+\left[\begin{array}{l} 0_{2 \times 1} \\ N \end{array}\right]r$$ My question: strangely enough, the pair $$(A_e,B_e)$$ is uncontrollable for the provided matrices, but it is controllable when I change to $$H = \left[\begin{array}{l} 1 & 0 \end{array}\right]$$. After designing a controller, it can track a ramp on the state $$q$$, which implies tracking a step on $$\dot{q}$$, but I feel that the former $$H$$ should yield identical behaviour when a step $$r$$ is applied to track $$\dot{q}$$. Is there an explanation for this?

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– AJN
Feb 15, 2022 at 15:24
• I ought to be able answer his but have forgotten my learning on matrices, sadly Feb 22, 2022 at 19:31
• The goal is the state $x_2$ to track a step input ? Feb 23, 2022 at 0:17

The uncontrollability makes sense, because the first error integral will be a shifted version of $$q$$, hence we cannot control them independently.