ELEC4632 Chap.2 Discrete-Time Models and Stability
Discrete-Time Models and Stability
Computer control works through sampled measurements and held inputs, so stability must be read from repeated state updates rather than from a continuous-time shortcut. This chapter interprets x(k+1) = Ax(k) + Bu(k), develops the role of a state-transition matrix, uses the tutorial matrix exponential and Cayley-Hamilton exercise, and classifies tutorial eigenvalues against the discrete-time unit circle.
What this chapter covers
- 01
State and input contributions to the next sampled state
- 02
Sampling, computation, input hold and measurement return
- 03
Autonomous evolution through repeated powers of the state matrix
- 04
Matrix exponentials and state-transition structure
- 05
Cayley-Hamilton reduction of higher matrix powers
- 06
Unit-circle classification of discrete-time eigenvalues
- 07
Distinction between mathematical stability and implementation quality
Worked example · free
Derive a matrix exponential from nilpotent structure
- +1The characteristic polynomial is (lambda+1) squared, giving a repeated eigenvalue at negative one.
- +1Write A = -I + N, where N squared equals zero.
- +1Use e to the power At = e to the power -t multiplied by (I + tN).
- +1At t equal to one, the result is e to the power -1 times the matrix with rows [2,-1] and [1,0].
Key terms
- Sample Index
- The sample index labels successive measurement and control updates in a discrete-time model.
- State Transition
- A state transition maps a current state into its future value under specified input conditions.
- Unit Circle
- The unit circle is the discrete-time boundary separating decaying eigenvalue magnitudes from growing ones.
- Cayley-Hamilton Theorem
- Cayley-Hamilton states that a square matrix satisfies its own characteristic polynomial.
- Nilpotent Matrix
- A nilpotent matrix becomes the zero matrix after a finite power, truncating related series.
Discrete-Time Models and Stability FAQ
Why is a negative pole not automatically stable in discrete time?
Discrete-time stability depends on eigenvalue magnitude rather than sign alone. A pole at negative one keeps its magnitude and alternates sign, while a pole below negative one in value can have magnitude above one and produce a growing oscillation.
What does Cayley-Hamilton help simplify?
Once the matrix identity is known, higher matrix powers can be rewritten using lower powers. That reduces repeated state-update calculations and supports later controller formulas that evaluate a desired characteristic polynomial at the state matrix.
Does a stable matrix guarantee good hardware performance?
No. Stability is necessary, but sampling rate, measurement noise, actuator saturation, offsets and modelling error still influence the real response. Simulation and implementation evidence are needed before a stable mathematical model becomes a defensible engineering design.
How should a pole on the unit circle be described?
Its magnitude does not decay under repeated updates, so it is not asymptotically stable. The exact response depends on multiplicity and matrix structure; avoid calling the system convergent merely because the pole is not outside the circle.
Why keep continuous and discrete stability rules separate?
Continuous-time decay is linked to eigenvalues with negative real parts, while sampled-state decay requires magnitudes below one. Writing the model domain before the test prevents a negative discrete pole with large magnitude from being classified by the wrong rule.
What should accompany a discrete stability classification?
State how repeated updates affect magnitude and, where useful, sign or phase. Then separate the mathematical classification from performance questions such as settling speed, disturbance rejection, sampling choice and control effort, which require additional evidence.
Exam move
Write the state equation in words before manipulating it: inherited state contribution plus input contribution. For each eigenvalue, state what repeated multiplication does to magnitude and sign. Redo the tutorial matrix exponential by separating the scalar and nilpotent parts, then verify the identity at zero time.
For Cayley-Hamilton, keep the scalar polynomial separate until its coefficients are complete, then substitute the matrix and check every entry. Create a stability table with columns for eigenvalue, magnitude, sign or angle, repeated-update behaviour and classification. Include a positive pole inside the unit circle, a negative pole inside it, a pole exactly on the boundary and one outside.
Sketch several samples so oscillation and growth are visible rather than memorised as labels. Next, practise moving between a state equation, its transition matrix and a verbal description of how the current state and input determine the next sample. For matrix exponentials, verify the value at zero and differentiate the expression to see whether it satisfies the governing relation.
When a continuous model is sampled, identify the sampling interval and zero-order-hold assumption before applying a discrete test. Finish every stability answer by separating mathematical classification from engineering performance: a stable model can still have slow response, excessive input, poor disturbance rejection or sensitivity to modelling error.
That distinction prepares the transition from analysis into controller design and hardware verification. Add a short domain check at the top of every practice answer: continuous or discrete, scalar or matrix, free response or forced response. This prevents a correct theorem from being applied to the wrong object.
For repeated updates, calculate two or three samples from a simple initial condition and compare them with the eigenvalue prediction. Finish by stating what further evidence, such as input demand or robustness, is still needed beyond stability.
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