Physical Significance of Stability, Controllability and Observability
This is state space model in form of 1st order linear ODE. In control, we find mathematical formula of any physical process in form of ODE and the beauty is any higher order ODE can be written in 1st order ODE in state space form.
Let’s see the significance of each variable in state space. Imagine a electric car as our system. “x” is state of system, ie variables which describe our physical state of our system (position, velocity). “u” act like an external actuator we use to alter the state of system (gas paddle, brake). “y” is output states, which we use to measure our state by sensor (speedometer, GPS). It is not necessary that “y” is equal to “x”, ie we do not have enough sensors (y) to measure each state (x) variables.
Matrix A, describes our dynamic model of car with its states(x). Matrix B is associated with input actuator(u). Matrix C, related to sensor output. By closed loop system, we alter the actual dynamic A, to new dynamic (A-BK). If A is unstable then (A-BK) may be stable. By choosing proper K matrix we force our eigen value of new system (A-BK) to achieve negative real part so that system is stable.
Definition of Stability:
The solution of linear ODE, contain a homogeneous (due to A matrix) and particular solution (due to B matrix).
Total response = Natural Response + Forced Response
For a useful system, generally natural response must goes to zero and leaving only forced response. If natural response will be greater than forced response then system will be unstable.
Another way is BIBO stability, that if we give limited and bounded control input(energy) then our output must be limited, measurable and bounded. The output should not be jump to infinity. We analyse what will happen to our states of system with time.
Controllability:
“By choosing proper “u” input, I can move to any “x” state”. Eg, [speed position] are two state of 1-D car, [Accelerator Brake] are two input. If the system is controllable, I can achieve any [speed position] by choosing right value of [Accelerator Brake]. Tell me a speed and position to go, I will use my accelerator and brake to reach there. It is controllable.
Sometime by one input, we can control more than one states, which is favorable to us. Or, sometime we are unable to directly control state “a”, but we control some other state “b” and might possible on changing “b”, “a” will also be change. Still system is controllable.
Why controllability is important ? First in design of stable system. Second, minimizing the number of control input. In above car example, our car has two input. But is there really need of two input ? Answer is No. We can remove our “Brake” and use only Accelerator to move or stop my car. Our car will be still controllable. Is it interesting ? Yes, In complex system (aeroplanes, missiles) we try to use only those minimum actuator to fly. If 8 actuators are enough, why to use 9th actuator ? That’s why we change wing design (A B matrix).
But here is a twist. “Controllability Gramian” tell us which state are more controllable or easy to move or by same input energy we can move the states most far away. In case, if we have a lot option, by which system is fully controllable, by help of Gramian we choose the best input. Point to be note, minimize the actuators but other factors like energy, robustness, stability (B matrix is in new dynamic) should not be compromised. Like, “removing the brake” means loosing the degree of controllability.
Observability:
“Do we have enough sensor measurement to know about all the state variables”. In above car example. our states are [speed position] and let say we have sensor [ speedometer encoder] to know my speed and position respectively. So both states are observable, speed can be measured by speedometer and position can be measured by encoder.
Why observability is important ? First in the controller design of a closed loop feedback system. Second minimizing number of output sensors. For example, in above car example, is it important to use both sensor ? No. Because we can either use Speedometer or Encoder. If we are using speedometer then we can integrate speed to find position. If we are using encoder, we can differentiate position to find speed. In both case, by using only one sensor we can measure our both state and it will be observable system. We may not have sensor to measure each physical states. We may not have enough money to buy two sensor. So observability help in this.
“Observability Gramain” tell us the order of most observable to least observable states. Some states “x” are easier to observe given the state of “y” or some states having less noise measurement than other. Example in the car example, which one to choose, Encoder or speedometer if both are giving me a observable system.
Further application: Only when a system is fully stable, controllable and observable, we can design a Full Estimator (LQE aka Kalman Filter) in which we estimate all states.
Now, lets see 3 definitions which are most practical condition in real world.
Stabilizability : A system is stabilizable if uncontrollable states are stable.
Detectability : A system is detectable if unobservable states are stable.
Duality : The pair (A, B) is controllable if and only if the pair (A , C) is observable.
Conclusion: Controllability and Observability help us in Actuator and Sensor selection, so that in minimum components we can gain maximum stability and less noisy system.
