Static State Feedback

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Capitolo 0. INTRODUCTION 4.1

Static State Feedback

• Control law:

– Open loop control: the control law is predetermined and it is a function of the initial state, final state and the coefficients of the system model.

– Feedback control: the control law takes into account, moment by moment, the evolution of the system.

• The feedback control law can have two different forms:

– Dynamic feedback: the control law is a “dynamic” function of the system state vector, that is the control unit itself has a dynamic behavior.

– Static feedback: the control law is a “static” function of the state vector of the system.

• With regard to the control goals we can distinguish between:

– Regulation problems: the reference signal is constant and the control action if designed to force the system trajectories towards the final asymptotically stable point also in presence of external disturbances acting on the system.

– Tracking problems: the input reference signal is time-varying and the control action if designed to force output signal to track the input signal in the best possible way with the minimum tracking error.

The design methods presented in the following sections will refer to the case of designing a static feedback control law to solve the regulation problem.

Zanasi Roberto - System Theory. A.A. 2015/2016

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Capitolo 4. REACHABILITY AND CONTROLLABILITY 4.2

Static state feedback

• Let us consider an invariant linear system S = (A, B, C, D) (discrete o continuous-time) and let us suppose that all the components of the state vector x are known.

˙x = Ax + Bu y = Cx + Du

u S y

x

• The system (in this case a continuous-time system) can be graphically represented in the following way:

u B

D

˙x s⁻¹ x

C y

A

x

S

• The static state feedback control law has the following form:

u(t) = K x(t)+v(t)

v u

S K

y x

where K ∈ R^m×n is the design a matrix and v(t) is un additional input.

• Applying the control law u = K x + v to the previous continuous-time system (the same holds for a discrete-time system) one obtains:

˙x = (A + BK)x + Bv y = (C + DK)x + Dv

v S_K y

x

The new dynamic system SK characterized by the following matrices:

A+ B K, B, C + D K, D.

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Capitolo 4. REACHABILITY AND CONTROLLABILITY 4.3

• The feedback system SK can be represented graphically as follows:

v B

D

˙x s⁻¹ x

C + DK y

A + BK S_K

• Property (Invariance of the reachability subspace). The reachability subspace XS⁺

K of the feedback system SK is equal to the reachability subspace X_S⁺ of the system S.

∀ K ∈ R^m×n ⇒ X_S⁺ = XS⁺

K

• Property (Invariance of the eigenvalues of the not reachable subsystem).

If the static state feedback control law u = K x = [K¹, K2] x is applied to a system S in the reachability standard form, one obtains a feedback system SK which itself is in the reachability standard form:

A = A1,1 A1,2

0 A2,2

, A + B K = A1,1 + B¹K1 A1,2 + B¹K2

0 A2,2

In both cases A²,2 is the matrix that characterizes the not reachable part of the system.

• Let us now consider the case of a control law obtained using a static feedback of the output signal y(t):

u(t) = K¹y(t) + v(t)

• For SISO systems (m = p = 1) matrix K¹ is a scalar parameter and therefore only one degrees of freedom can be used to modify the dynamics of the feedback system. On the contrary, when a state feedback control is used matrix K is a vector of dimension n and therefore n degrees of freedom can be used to modify the dynamics of the corresponding feedback system.