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This chapter describes how feedback can be used shape the local behavior of a system. The concept of reachability is introduced and used to investigate how to "design" the dynamics of a system through placement of its eigenvalues. In particular, it will be shown that under certain conditions it is possible to assign the system eigenvalues to arbitrary values by appropriate feedback of the system state.
This chapter describes how state feedback can be used to design the (closed loop) dynamics of the system:
A linear system with dynamics
is said to be reachable if we can find an input defined on the interval that can steer the system from a given final point to a desired final point .
The reachability matrix for a linear system is given by
A linear system is reachable if and only if the reachability matrix is invertible (assuming a single intput/single output system). Systems that are not reachable have states that are constrained to have a fixed relationship with each other.
A linear system of the form
is said to be in reachable canonical form. A system in this form is always reachable and has a characteristic polynomial given by
A reachable linear system can be transformed into reachable canonical form through the use of a coordinate transformation .
A state feedback law has the form
where is the reference value for the output. The closed loop dynamics for the system are given by
The stability of the system is determined by the stability of the matrix . The equilibrium point and steady state output (assuming the systems is stable) are given by
If a system is reachable, then there exists a feedback law of the form
the gives a closed loop system with an arbitrary characteristic polynomial. Hence the eigenvalues of a reachable linear system can be placed arbitrarily through the use of an appropriate feedback control law.
Integral feedback can be used to provide zero steady state error instead of careful calibration of the gain . An integral feedback controller has the form
is the integral error. The gains , and can be found by designing a stabilizing state feedback for the system dynamics augmented by the integrator dynamics.
A linear quadratic regulator minimizes the cost function
The solution to the LQR problem is given by a linear control law of the form
where is a positive definite, symmetric matrix that satisfies the equation
This equation is called thealgebraic Riccati equation and can be solved numerically.
Frequently Asked Questions
- Errata: In Exercises 2.4 and 5.9, the (2,1) entry of the dynamics matrix should be ab-b
- Errata: In heuristic derivation of reachability test, α is missing subscript
- Errata: In equation (6.5), the 4, 3 entry of W r should be g m^2 l^2 M t/mu
- Errata: In Example 6.3, z 2 should be x 1/omega instead of x 2/omega
- Errata: In equation (6.17), elements of the output matrix should be in the opposite order
- Errata: Reference to equation (6.17) should be to equation (6.15)
- Errata: Formula for k r in equation (6.21) of Theorem 6.3 is incorrect
- Errata: Missing 's' in expression for characteristic polynomial in discussion of eigenvalue assignment
- Errata: In Example 6.5, definitions of z1 and z2 are switched
- Errata: In Example 6.5, reference variable should be L d instead of r
- Errata: In equation (6.22), k omega^2 should be k omega 0^2
- Errata: In the caption for Figure 6.7, L e=20 should be L e=30
- Errata: For underdamped second order system, eigenvalues are listed incorrectly
- Errata: In equation (6.24), the sign of the sin(omega t) term is incorrect for zeta less than one
- Errata: Values of zeta in the caption for Figure 6.8 don't match the figure
- Errata: In Example 6.7, the imaginary part of the first pair of eigenvalues is missing factor of i
- Errata: In Example 6.7, the open loop eigenvalues are incorrect
- Errata: Numerical gains for Example 6.7 are incorrect
- Errata: In Example 6.8, the state space dynamics have errors in the trigonometric terms
- Errata: Diagonal entries for Q v should be rho
- Errata: In Example 6.8, the state error should be \xi and the input should be F
- Errata: The formula given for the rise time of a second order system in Table 6.1 is only approximate
- Errata: In Example 6.10, the reference gain should be k r = a 1/b
- Errata: In Example 6.10, the system input for the linearized model should be w instead of u
- Errata: In Exercise 6.4, need to assume A is invertible and C A^-1 B is nonzero
- Errata: In Exercise 6.13, R should be removed and rho should be rho 1
- Errata: In Example 6.6 and 7.1, the output equation has 'x' for the state instead of 'c'