Control design with guaranteed transient performance: An approach with polyhedral target tubes

Abstract

In this paper a novel approach is presented for control design with guaranteed transient performance for multiple-input multiple-output discrete-time linear polytopic difference inclusions. We establish a theorem that gives necessary and sufficient conditions for the state to evolve from one polyhedral subset of the state-space to another. Then we present an algorithm which constructs a time-varying output feedback law which guarantees that the state evolves within a time-varying polyhedral targettube specifying the system's desired transient performance. We present generalisations involving constraints on the control, and a bounded additive disturbance term. Our formulation is very general and includes reference tracking with any desired transient behaviour in the face of disturbances, as specified, for example, by the most popular step response specifications. The approach is demonstrated by an example involving the control of water levels in two coupled tanks.

Prescribed performance tracking for a class of uncertain nonlinear systems with actuator failures

“…From the previous investigation, we know that there are four methodologies used to achieve the specified transient and steady state performance, that is, adaptive control schemes, [11][12][13] the funnel control scheme, [14][15][16][17] the PPC scheme, [18][19][20][21][22][23] and the target-tube approach. [24][25][26] Especially in PPC, [18][19][20][21][22][23] the asymptotic performance function and the finite-time performance function (FTPF) are used to generate funnel-shaped decaying bounds. However, the transient and steady-state performance results can only be achieved in the above four methodologies according to certain preassigned bound forms.…”

“…The new prescribed performance tool has the following innovative features: (1) it offers a new idea for solving the finite-time problem, and this is obviously better than asymptotically achieving a specified performance; [18][19][20][21][22] (2) the first-order derivative of the FTPF in the work of Liu et al 23 needs to be less than 0, and the output of the chained first-order filters in (1) has no such restriction; (3) with the aid of the new prescribed performance tool, when sufficiently small values of 𝜏 g , g = 1, 2, … , s and T are chosen, any preset continuous bounds can be formed. In other words, the bounds of the transient performance can be prescribed in an arbitrary way with flexibility and adaptability; this is obviously better than the techniques used in the literatures; [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] 6) through the analyses of ( 4) and ( 5), the application of this new prescribed performance tool is more convenient than the prescribed performance methodologies involved in the literatures [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] because the s...…”

“…Consider the closed-loop uncertain nonlinear system in (7) with the presence of actuator failures in (8) under Assumptions 1-4. The stabilizing signal in (21), (22), and the actual controller in (23) with parameter update laws ( 24)- (26) can ensure that (1) the tracking error signal z 1 converges in the mode determined by the set G and (2) all signals in the closed-loop system are bounded.…”

“…For further details of polyhedral target tubes, the reader is referred to the work of Esterhuizen and Wang. 26 A common feature in the previous works [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] is that the bounds of the transient performance cannot be prescribed by utilizing an arbitrary design. Apart from this, the transient performance of a model that contains actuator failures has not drawn any attention in the aforementioned literature.…”

“…Compared with the existing works on guaranteed transient and steady state performance. [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] The new prescribed performance tool proposed by us possesses better features in terms of flexibility, adaptability, and convenience. First, by choosing the prescribed reference coordinate values, sufficiently small coordinate value intervals and the gain parameters of the chained first-order filters, any preset continuous bound can be generated; that is, the bounds of the transient performance can be prescribed in an arbitrary way.…”